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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)

#51

Post by Koshkin »

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.

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…

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

#52

Post by eyelidlessness »

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...

My lofty future academic pursuit for fun when I can relax from my professional programming career would probably be compsci and then research if I’m still interested. Gonna get really old and tired working in the semicolon mines, then turn around and learn all the stuff I was supposed to learn when I started. At least that’s my picture of my casual retirement interests.

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

#54

Post by l33t2328 »

Koshkin wrote: post #51
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.

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.

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

#55

Post by ivansavz »

hebrox wrote: post #15
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...

I'm actually looking into this. Just an hour ago I mailed the local university that I won't be doing any courses there. My initial plan was to do a bachelor at around 50% speed. But working and having to girls (1,5 years and 3 weeks) makes that quite impossible. And looking at photos of myself at 17 makes me feel rather out of place at a university at age 42. The Open University has an AI master that I'm thinking abo…

RE learning math: you might want to check out some of my (non-free) books on MECH+CALC https://minireference.com/static/excerpts/noBSmathphys_v5_pr... and LINEAR ALGEBRA https://minireference.com/static/excerpts/noBSLA_v2_preview.... They are written especially with adult readers in mind.

RE schedule to keep you going: I've had some students use the concept maps as a "world map" (like the stage map in Mario World) and check off boxes as you progress through them (each concept corresponds to roughly one section). See https://minireference.com/static/conceptmaps/math_and_physic... and https://minireference.com/static/conceptmaps/linear_algebra_... I guess you could time-box these and do N of sections each week to make this into a schedule. Make sure you dedicate lots of time for the exercises/problems, because that's when the real learning happens...

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

#56

Post by l33t2328 »

Koshkin wrote: post #53
l33t2328 wrote: post #43

Earlier quoted context omitted.

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

But the real numbers cannot be faithfully represented in a computer!

So? Neither can the list of possible computer programs, yet CS studies them

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

#58

Post by ravi-delia »

l33t2328 wrote: post #43
qiskit wrote: post #13

Earlier quoted context omitted.

CS is math...

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 and CS are set equal, but one predefines an entry point and the other does not. From the human perspective math gives rise to CS, which is just one of math's many children.

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

#59

Post by ravi-delia »

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 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.

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

#60

Post by 07121941 »

nextos wrote: post #37
paulpauper wrote: post #5

Earlier quoted context omitted.

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…

I totally agree with what you say, but there's a lot of low hanging fruit in mathematizing biology. It's not easy, but it's certainly very impactful.

can you expand on that? sounds really interesting
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