Edward Frenkel
speaker
2,304 appearances
1 recordings
1 series
first heard Apr 2023
last heard Apr 2023
Edward Frenkel’s voice in public audio — every appearance, attributed to the second.
Trend
recordings per month · last 12 monthsNo recordings in the last 12 months.Older appearances are listed below; set an alert to hear about the next one.
Appearances
And then the idea that was basically prevalent in the world of mathematics by the beginning of the 20th century was that in principle, all of mathematics could be derived this way.
We just have to find the correct system of axioms, and then everything you ever need could be produced by this procedure, which is really algorithmic procedure, which actually could be run on a computer.
Now, think about it.
What is special about this process?
In this process, you are just manipulating symbols, basically.
You're going from one statement to another without really understanding the meaning of it.
So it's an ideal playground for a computer program.
It's a purely syntactic process where there are some rigid rules of passing from one statement to the next.
Most mathematicians believed that this way you can produce all true statements.
And if this were true, it would give a lot of credibility to the thesis that everything in life is computational.
Or life is computation.
Because then, at least mathematics is computational because then it can be programmed.
And a computer, after sufficient time, depending on its capacity, would produce every true statement.
So, Gödel's first incompleteness theorem says that that's not the case.
It not just says it, but it proves it at the highest level of rigor that is available in mathematics.
That is to say within another formal system that he was operating in.
More precisely, what he proved was that if you have a sufficiently sophisticated formal system, that is to say that you can talk about numbers, whole numbers in it, that you have whole numbers, one, two, three, four, you have formalized the operation of addition and multiplication within the system,
If it is consistent, that is to say, if it's not completely useless, then there will be a true statement in it, which cannot be derived by this linear syntactic process of proving theorems from axioms.
It's really incredible.
So this was a revolution, 1931, revolution in logic, revolution in mathematics, and we're still feeling the tremors of this discovery.
Showing 961–980 of 2,304 · page 49 of 116
← Previous
Next →