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Mathematicians are chronically lost and confused (2014)

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Re: Mathematicians are chronically lost and confused (2014)

#41

Post by throwawaysocks »

stepvhen wrote: post #38
laingc wrote: post #13

To me, this is about "mathematical maturity". My observation is that many programmers, especially those who have come of age by working in startups, tend to value ability and sometimes experience over formal education. This is a result, I believe, of noticing that they can outperform many people who have a classical education, and also seeing that many of the people to whom they look up also do not have much in the w…

I think math is "easier" to learn while it still holds a clear practical value, up to calculus and linear algebra. Past that, when you start to enter the world of "pure" mathematical s, it can be muh more difficult. The practical value of earlier subjects allows a student or what have you to draw connections between what they already know and this new concept. Something like group theory however is more difficult to…

My experience was the opposite. The more practical Mathematics was too easy to hand wave through so no one ever bothers with proof. But then when something goes wrong or you're not sure whether a particularly complicated calculation makes sense, you're SOL without a real formalism to go on.

I didn't really believe in calculus except on a case by case basis until after my first analysis course.

I think the only reason the less formal mathematics courses seem easier is because the thing being studied is also dead simple and the calculations are easy. But that approach doesn't scale.

Re: Mathematicians are chronically lost and confused (2014)

#42

Post by yequalsx »

agentultra wrote: post #30
laingc wrote: post #13

To me, this is about "mathematical maturity". My observation is that many programmers, especially those who have come of age by working in startups, tend to value ability and sometimes experience over formal education. This is a result, I believe, of noticing that they can outperform many people who have a classical education, and also seeing that many of the people to whom they look up also do not have much in the w…

I disagree that mathematics itself is difficult to engage with outside of academia. There are some academic mathematicians who think mathematics has to be hard because it was difficult for them to understand. There are others for whom sharing their insights is more important than searching for new ones. I think that the set of mathematicians together have done a rather good job in the 20th century of producing mathem…

Not knowing your level of mathematical insight and knowledge makes it hard to know if your belief about your mathematical talent is a self deception. I've never encountered anyone who understood typical second year graduate level mathematics without formal training. I know such people could exist. I've just never met any. I have met people who claimed to be self taught in mathematics and it was obvious that they didn't really understand the concepts though they were absolutely convinced they knew what they were talking about.

Outliers exist. But have any good mathematicians been self taught in the last 50 years?

Re: Mathematicians are chronically lost and confused (2014)

#43

Post by jondubois »

My favourite quote about Mathematics is from John von Neumann: "In mathematics you don't understand things. You just get used to them." - This quote highlights precisely why I ended up choosing software engineering over maths.

I'm just not very good at applying processes/methodologies which I don't fully understand.

For example, I wasn't very good with linear algebra until I was able to visualize the equations in my head. For example, now, when I think about the equation 'f(x) = ax^2 + bx + c' - I can see that this represents the set of all possible quadratic equations and I can roughly visualize what that looks like on a cartesian plane (well it would turn the whole plane black because there would be an infinite number of graphs). Then if I choose any three points on that crowded cartesian plane, I can visualize that among this infinite set of curves, one of them passes through all three points. Thinking about it in that way allows me to make sense of Gauss-Jordan Reduction and other mathematical processes related to linear algebra.

Programming is much easier for me because I can visualize the results instantly on a computer - I don't need someone else to explain it to me. Any uncertainty can be quickly resolved by simply running some code.

Re: Mathematicians are chronically lost and confused (2014)

#44

Post by pixl97 »

jondubois wrote: post #43

My favourite quote about Mathematics is from John von Neumann: "In mathematics you don't understand things. You just get used to them." - This quote highlights precisely why I ended up choosing software engineering over maths. I'm just not very good at applying processes/methodologies which I don't fully understand. For example, I wasn't very good with linear algebra until I was able to visualize the equations in my…

>Any uncertainty can be quickly resolved by simply running some code.

Until you have to interact with a black box of someone else's code. You can only be certain that the data you sent that particular time works, not that all possibilities of valid data work.

Re: Mathematicians are chronically lost and confused (2014)

#45

Post by pixl97 »

redcalx wrote: post #36
agentultra wrote: post #30

Earlier quoted context omitted.

I disagree that mathematics itself is difficult to engage with outside of academia. There are some academic mathematicians who think mathematics has to be hard because it was difficult for them to understand. There are others for whom sharing their insights is more important than searching for new ones. I think that the set of mathematicians together have done a rather good job in the 20th century of producing mathem…

> proof of weak fairness Interest piqued. Do you have a good link for an intro to this?

http://bluehawk.monmouth.edu/rclayton/web-pages/u03-598/slf....

Re: Mathematicians are chronically lost and confused (2014)

#46

Post by nextos »

pmiller2 wrote: post #40
nextos wrote: post #20

Earlier quoted context omitted.

I'd also recommend Hubbard & Hubbard for a beautiful and beginner-friendly mix of algebra and analysis. My preferred starter kit is Rudin plus Halmos or Axler, but treating Rudin as a summary. So a helper would be needed, like Counterexamples in Analysis. This is what Math 55 used to do.

I don't really recommend Rudin for a true beginner at all (unless, by "as a summary," you mean not really digging into the proofs themselves, in which case any good analysis book will do). Rudin will always try to take the most elegant route to the theorem, regardless if that route goes anywhere near where the rest of the text has been. The result, for me, has been that many of his proofs seem to just meander about f…

I should have said as an outline, instead of a summary.

I think the beauty of Rudin is how compact it is. But of course you need an alternative book to be able to digest it.

Re: Mathematicians are chronically lost and confused (2014)

#47

Post by stuxnet79 »

tgb wrote: post #27
505 wrote: post #14

Earlier quoted context omitted.

Since philosophy was your favourite, you might enjoy the novels of Greg Egan. As far as I can tell, he puts good mathematics in his stories.

I recently read my first Greg Egan and enjoyed it (Quarantine). Got any recommendations for specifically the ones he put good math into?

Schild's Ladder is terrific.

Re: Mathematicians are chronically lost and confused (2014)

#48

Post by marcosdumay »

vecter wrote: post #16

I have a great personal story that highlights how long the journey of understanding mathematics is. I took linear algebra my freshman year of college. It was the non-math major course, so it didn't require proofs. I got an A+ in the class. Not just an A, an A+. I was able to obtain such a high grade by taking tons of practice tests, and since the actual tests were basically mildly veiled calculations, I just had to m…

In general, mathematicians are bad teachers for introductory Math courses (shouldn't be very surprising, experts are often bad teachers for introductory courses). Too bad universities think they are the most prepared for that.

I had a similar experience. Not only I passed linear algebra without understanding linear spaces, but I passed analytic geometry without learning any geometric concept that I didn't know.

Re: Mathematicians are chronically lost and confused (2014)

#49

Post by jondubois »

pixl97 wrote: post #44
jondubois wrote: post #43

My favourite quote about Mathematics is from John von Neumann: "In mathematics you don't understand things. You just get used to them." - This quote highlights precisely why I ended up choosing software engineering over maths. I'm just not very good at applying processes/methodologies which I don't fully understand. For example, I wasn't very good with linear algebra until I was able to visualize the equations in my…

>Any uncertainty can be quickly resolved by simply running some code. Until you have to interact with a black box of someone else's code. You can only be certain that the data you sent that particular time works, not that all possibilities of valid data work.

I guess with programming, you can never get 100% certainty when it comes to correctness of your code - Unless you perform formal analysis which is impractical for most use cases.

Re: Mathematicians are chronically lost and confused (2014)

#50

Post by larme »

vecter wrote: post #16

I have a great personal story that highlights how long the journey of understanding mathematics is. I took linear algebra my freshman year of college. It was the non-math major course, so it didn't require proofs. I got an A+ in the class. Not just an A, an A+. I was able to obtain such a high grade by taking tons of practice tests, and since the actual tests were basically mildly veiled calculations, I just had to m…

Just out of curiosity, do you remember the title of the PCA paper?
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