I'm still grumpy that I accepted I knew what real numbers were just because I could recite back the definition given by the teacher. There is so much depth there if you go looking...
I contend no one understands real numbers if they can’t explain (without resorting to essentially purely symbolic rigor) why you can cover the rationals with intervals of arbitrarily small total length, but you can’t do the same with R.
Well, first you would have to explain what you mean by your statement.
You can infer from this how Artificial Intelligence and Logic should be combined: AI enables a "post-rigorous" mode, and Logic is how you know you are still doing something sensible, and how you expand your sure footing.
One day I'll retire and go back to school. The idea of learning Math - really learning & understanding Math - as a fun pastime is so appealing. What's stopping me now? That sweet overpaid SDE salary and the endless obligations that come from being an adult. I suspect I am not alone...
keep the salary. if the goal it to change the world, to understand reality, to have an impactful life, have a good standard of living, etc. math is one of the hardest ways of achieving that. It's such a saturated field. Almost everything you can imagine has been done to the highest possible degree of abstraction. Every stone overturned except for things which may take a lifetime to even try to understand. Writing a b…
just a minor thing to note here: there's the foundations of mathematics... questions like, what are numbers and other mathematical structures? How is it that math, any math at all, can potentially accurately describe parts of reality? Symbolic representations of math are just squiggily lines... why should squiggily lines have any special relationships to the nature of the universe? Details like that are far from being understood.
keep the salary. if the goal it to change the world, to understand reality, to have an impactful life, have a good standard of living, etc. math is one of the hardest ways of achieving that. It's such a saturated field. Almost everything you can imagine has been done to the highest possible degree of abstraction. Every stone overturned except for things which may take a lifetime to even try to understand. Writing a b…
just a minor thing to note here: there's the foundations of mathematics... questions like, what are numbers and other mathematical structures? How is it that math, any math at all, can potentially accurately describe parts of reality? Symbolic representations of math are just squiggily lines... why should squiggily lines have any special relationships to the nature of the universe? Details like that are far from bein…
maybe I'm misunderstanding you, it sounds like you're just describing abstract algebra. See linear algebra or group theory as topic titles.
Well, to understand an explanation one must already know something, otherwise first you have to explain those other things, e.g. the simple fact that unlike the reals the set of rational numbers is countable…
Cardinality hasn’t much to do with it since there are uncountable sets which you may cover like that.
Sure, but for countable subsets (such as the rationals) it is easy to show.
That is not at all what the incompleteness theorems say. The first incompleteness theorem says that for any consistent formal system T (with a recursively enumerable set of axioms) capable expressing of elementary arithmetic, T can express a statement which it can neither prove nor disprove. The second incompleteness theorem says that T can't prove the statement "T is consistent". (I've still glossed over a number of…
Yes, it is. Sorry if you can’t read deeply into it or something. I’m not posting for grad students. I can sense you just like to correct people. Ahhhh I’m so wrong, you’re right?
Godel's theorems mean something quite specific, and rely on an equally specific set of hypothesis.
It's tempting to try to apply them (or rather the same kind of conclusions) in other (non math) contexts, but it's very not obvious that you'll get something sensible. While you can play with the ideas, invoking Godel's theorem outside of its specific context doesn't make much sense.
I contend no one understands real numbers if they can’t explain (without resorting to essentially purely symbolic rigor) why you can cover the rationals with intervals of arbitrarily small total length, but you can’t do the same with R.
Well, first you would have to explain what you mean by your statement.
I mean that for any epsilon > 0, you can have a set of intervals of the form (a_i, b_i) where every rational number is in some interval and the sum over all i of b_i - a_i That is, you can cover the rationals with intervals of arbitrarily small total length.
Even if no one remembers you, your work is a contribution. Learning for its own sake is a hobby for yourself, like watching TV or reading a book.
Let me be remembered for watching TV. I would be utterly delighted if that’s my long term legacy. “They finally rested and enjoyed the most banal show imaginable. It was relaxing. Any other lasting contributions will be in other records should you care.” There I’ve written my eulogy.
My grandfather's last 10 years were spent developing diabetes, watching those three shows, and playing Microsoft solitaire until death. If that's what you want, then, sure, you can be remembered that way.
I've been thinking about the reals a lot...beyond the rationals are all the real numbers like pi that have finite definitions, (even if those definitions, like pi's, require infinite computation.) But there is also this vast set of reals that are simply undefineable, non-repeating sequences. These numbers are unmentionable and unknowable. Does it really even make sense to say that this subset of the reals exists in t…
Yes, of course. Because definable (really, computable) numbers don’t really “exist” either.
We can can use definable numbers. Pi is used with such massive frequency that is seems silly to say it has the same nonexistence as numbers that we can literally never even mention, let alone use.
My assertion is that there is something wonky with these u definable numbers and that wonkiness is directly related to how absolutely massive the infinity of reals is.