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There’s more to mathematics than rigour and proofs (2007)

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Re: There’s more to mathematics than rigour and proofs (2007)

#61

Post by l33t2328 »

ravi-delia wrote: post #58
l33t2328 wrote: post #43

Earlier quoted context omitted.

I’d argue math is actually (a proper subset of) CS.

Math is at it's broadest the study of formal systems. Computer science is the study of a particular formal system. While it is a powerful enough system to contain all of math within it, there are many such systems nested within each other. Is the Turing machine formalism more powerful? No. Is it more efficient or intuitive? Also no. It requires axiomatic reasoning to construct, and then reproduces it internally. Math…

I’d argue that logic is a subfield of computability theory, and so by extension we (in CS) absorb math.

Re: There’s more to mathematics than rigour and proofs (2007)

#62

Post by ifokiedoke »

devnulll wrote: post #2

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...

Same situation, but I've started! I only spend about 30 minutes every other day, and it's extremely slow-going, but so fulfilling. I had the same goal as you -- _really_ learning & understanding math.

I finished pre-Algebra last year and I'm halfway through an Algebra text by Gelfand & Shen now. My friends look at me funny when I tell them I'm re-learning Math from the ground up for fun (esp. with a degree in CS) but it has been so rewarding. I probably won't get to finishing Calculus for another couple years but I'm already having so much fun. Stumbled upon deriving some exponent laws last month by accident and truly understanding the sum and difference of squares has been awesome.

Re: There’s more to mathematics than rigour and proofs (2007)

#63

Post by hither_shores »

postingposts wrote: post #48
andi999 wrote: post #16

Earlier quoted context omitted.

About the simulation hypothesia, how is that different from Descartes evil demon? https://en.m.wikipedia.org/wiki/Evil_demon (apart from saying that the evil demon is a future type computer)

The simulation stuff is just “Plato’s Cave”:Reloaded for people who never understood the concept of the cave in the first place. At a basic level you can interpret it from Gödel’s incompleteness theorems, which state that systems of logic need some form of observer to function. That observer concept reaches way back across different philosophical domains and authors as well.

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 technical details here; pick up a book on model theory if you want all the messy internals.)

First order logic is notably not capable of expressing elementary arithmetic. And observers aren't involved in any way.

Re: There’s more to mathematics than rigour and proofs (2007)

#64

Post by [deleted] »

ravi-delia wrote: post #59
l33t2328 wrote: post #42

Earlier quoted context omitted.

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.

I feel like that's downstream of seeing the reals as the rationals with the holes plugged, no? Obviously you can start from either end but everything special about the reals comes from being the complete version of the rationals.

[deleted]

Re: There’s more to mathematics than rigour and proofs (2007)

#65

Post by eyelidlessness »

lupire wrote: post #40
bombcar wrote: post #10

Earlier quoted context omitted.

I think the whole point would be to retire and do math for fun, not desiring anything more than the joy of discovery, no footnotes, no recognition, just math. 99.99% of everyone won't be remembered for their "contributions" so why not do something you enjoy?

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.

Re: There’s more to mathematics than rigour and proofs (2007)

#66

Post by shkkmo »

l33t2328 wrote: post #42
foobarian wrote: post #6

Earlier quoted context omitted.

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.

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 the same way that definable numbers do?

Re: There’s more to mathematics than rigour and proofs (2007)

#67

Post by l33t2328 »

shkkmo wrote: post #66
l33t2328 wrote: post #42

Earlier quoted context omitted.

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.

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.

Re: There’s more to mathematics than rigour and proofs (2007)

#68

Post by postingposts »

hither_shores wrote: post #63
postingposts wrote: post #48

Earlier quoted context omitted.

The simulation stuff is just “Plato’s Cave”:Reloaded for people who never understood the concept of the cave in the first place. At a basic level you can interpret it from Gödel’s incompleteness theorems, which state that systems of logic need some form of observer to function. That observer concept reaches way back across different philosophical domains and authors as well.

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?

Re: There’s more to mathematics than rigour and proofs (2007)

#69

Post by MichaelBurge »

postingposts wrote: post #68
hither_shores wrote: post #63

Earlier quoted context omitted.

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?

I usually think of logics as search algorithms, since that's how the semantics for the meta-language describing them are defined.

I guess you could call a Turing Machine implementing the search algorithm for proofs implied by a logic an "observer", since it produces "reachability observations" i.e. proofs.

Re: There’s more to mathematics than rigour and proofs (2007)

#70

Post by yobbo »

devnulll wrote: post #2

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...

If your goal of learning is for your own benefit, it's much easier than actually going to school and doing exams, assignments, and getting credits which involves a massive amount of formalities.

A good starting point could be MIT OCW 18.01, 18.02, and 18.03. Do all their problem sets and get as much understanding you want. It corresponds to a first year university engineering curriculum.

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