/sci/ - Science & Math

>>6078613
>>6078636
Godel's incompleteness theorems do NOT imply that mathematics is inconsistent; you are scolding him for understanding something that you can't. It's unlikely that you've ever even seen the proof, or you'd not make such a grave mistake. I thus conclude that you are just a Godel-Escher-Bach-lover who skims through articles and blog-posts, understanding nothing of what is being said. You are a pretender.

Allow me to explain the nature of Godel's completeness and incompleteness theorems. First of all, the completeness theorem says that a consistent first-order theory has a model - a universe of sets satisfying the axioms. It also says that you can then prove every formula that is true in ALL models.
The first incompleteness theorem says that every model of a consistent theory which contains Peano arithmetic includes undecidable propositions - statements that you cannot prove from the axioms because they are true in some models and false in others. People sometimes state this as "there are true statements that cannot be proven", but that relies on the idea that mathematical theorems are either true or false objectively, whether they can be derived from the axioms or not (the Platonic view). I prefer to see it differently. If you think of the axioms of mathematics as comparable to the axioms of, say, groups then the analogy would be in the statement: "the group is abelian: ab = ba". This cannot be proven true, nor proven false, because some groups (models of the group axioms) are abelian and others are not.
The second incompleteness theorem goes further and gives an example of a statement which cannot be proven true by a consistent theory. That is the statement "this theory is consistent". Combining this with the completeness theorem of earlier, we see that there must be a model of our theory in which the statement "this theory is consistent" is false!

It is possible for maths to be consistent - he just can't prove it!

G.H. Hardy once said something along the lines of "a mathematicians most developmental period is between 18 and 25".

I decided to take a break after high school and now I'm just starting college at 21.

Did I screw myself out of being a good mathematician?

>>5912021
College must be difficult for you.

"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."

http://www.math.ualberta.ca/~mss/misc/A%20Mathematician's%20Apology.pdf

I have only seen Lockharts lament here many times, but not Hardys piece. So here is link. Thoughts on this? I wonder what he would have to say about modern use of number theory in cryptography.

>>4925969
>non-standard analysis
>advanced
pick exactly one

>>3964602
G H Hardy, one of the greatest pure mathematicians ever, worked on evolution.

http://en.wikipedia.org/wiki/Hardy%E2%80%93Weinberg_principle

"No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." - Godfrey Harold “G. H.” Hardy

Discuss.

ITT: Badass mathematician quotes.

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." - G.H. Hardy

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." - G.H. Hardy

ITT: Badass Mathematician quotes:

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." - G.H. Hardy