161–170 of 199 posts
That PDF is just a bunch of scanned images of the book. It's large and cumbersome in many readers. There is a much better PDF at Project Gutenberg [1]. The Gutenberg PDF is only 1.9 MB, compared to 12 MB for the scanned image PDF. The Gutenberg page for this book [2] also has a link to the LaTeX source for the PDF. [1] http://www.gutenberg.org/files/33283/33283-pdf.pdf [2] http://www.gutenberg.org/ebooks/33283
I'm embarassed somewhat to say this, but over the past few weeks I've been taking the courses on Khan Academy on mathematics. I'm nearly 30. and I'm not talking about brushing up on my linear algebra, that comes later, I'm talking high school level mathematics, stuff that I've largely forgotten or didn't "get" first time round. I've seen these "machine learning for hackers!" articles who try to dish out a bit of math…
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Aren't metaphors and analogies representations that allow us to extract patterns?
To me, metaphors and analogies are about trying to learn some new concept(s) in terms of already known and familiar concepts.
From "Real Mathematical Analysis" 1st edition, p. 9:
Metaphor and Analogy
In high school English, you are taught that a metaphor is a figure of speech in which one idea or word is substituted for another to suggest a likeness or similarity. This can occur very simply as in "The ship plows the sea." Or it can be less direct, as in "his lawyers dropped the ball." What gives a metaphor its power and pleasure are the secondary suggestions of similarity. Not only did the lawyers make a mistake, but it was their own fault, and, like an athlete who has dropped a ball, they could not follow through with their next legal action. A secondary implication is that their enterprise was just a game.
Often a metaphor associates something abstract to something concrete, as "Life is a journey." The preservation of inference from the concrete to the abstract in this metaphor suggests that like a journey, life has a beginning and an end, it progresses in one direction, it may have stops and detours, ups and downs, etc. The beauty of a metaphor is that hidden in a simple sentence like "Life is a journey" lurk a great many parallels, waiting to be uncovered by the thoughtful mind.
Metaphorical thinking pervades mathematics to a remarkable degree. It is often reflected in the language mathematics choose to define new concepts. In his construction of the system of real numbers, Dedekind could have referred to A|B as a "type-two, order preserving equivalence class", or worse, whereas "cut" is the right metaphor. It corresponds closely to one's physical intuition about the real line. See Figure 3. In his book, Where Mathematics Comes From, George Lakoff gives a comprehensive view of metaphor in mathematics.
An analogy is a shallow form of metaphor. It just asserts that two things are similar. Although simple, analogies can be a great help in accepting abstract concepts. When you travel from home to school, at first you are closer to home, and then you are closer to school. Somewhere there is a halfway stage in your journey. You know this, long before you study mathematics. So when a curve connects two points in a metric space (Chapter 2), you should expect that as a point "travels along the curve," somewhere it will be equidistant between the curve's endpoints. Reasoning by analogy is also referred to as "intuitive reasoning."
Moral: Try to translate what you know of the real world to guess what is true in mathematics.
MIT recorded a set of Calculus video courses back in 1970s that they have since made publicly available. It is taught by a lecturer named Herbert Gross. His style of lecturing is clear, he states why things are defined the way they are and derives everything from first principles. There is an unusual mix of rigor and focus on building understanding - where everything comes from. It also taught me that math is about r…
It was still entirely relevant to today even though it was a few decades old as the fundamentals of computer science are still fundamental.
https://www.youtube.com/watch?v=2Op3QLzMgSY&list=PL8FE88AA54...
Hearing that intro music still brings a smile to my face.
I just happen to be relearning math right now as I dive deeper into data science and this is perfect timing. Going to watch this series once I get through my math proofs book ("Book of Proof" by Richard Hammack which I recommend to people getting into math https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472...).
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> Why didn't my professors do that? Because to them it's obvious. Most maths teachers are so far ahead of the students they forgot they once were students themselves. I've had 3 different ones in high school and the difference was incredible. All the way from 'only the best students learn anything' to 'everybody earns at least a passing grade'. Maths and physics were the classes where the quality difference between t…
> Because to them it's obvious. And to them it's wrong! Much is said in this book which is difficult (but not impossible) to rigourously justify. It took centuries for calculus to be placed on a rigourous mathematical foundation; this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. The presentation here (in particular the i…
Thank you for posting this! Such things always bothered me in high school, seemed like approximations that ought to bite you in the behind at least in some corner cases. Another example from TLA:
> dy = 2cos(θ + 1/2 dθ) · sin 1/2 dθ
> But if we regard dθ as indefinitely small, then in the limit we may neglect 1/2 dθ by comparison with θ, and may also take sin 1/2 dθ as being the same as 1/2 dθ. The equation then becomes:
> dy = 2cosθ × 1/2 dθ
This again seems like very sloppy and careless kind of approximation that ought to bite you in the back - but knowing there are just (supposed-to-be) intuitive non-rigorous methods, and that these have actual rigorous backing, somehow soothes me.
Summer of 1980, going into my senior year in high school, I mentioned I'd be taking Calculus next year to a co-worker a couple years older than I. He said he had the best book in the world on Calculus, and he loaned me his copy of Silvanus P Thompson's Calculus made easy . I thoroughly enjoyed that book, benefited from its intuitive explanations, and forever appreciated his recommendation. If I may similarly influenc…
Thanks so much for the kind words. I vividly remember cramming for a test my freshman year of college, not having things click, and the final Aha! when a semester of pain disappeared with a few visualizations. The contrast between how most classes presented the material and what actually worked for me was jarring, and I had to share what helped. I hope other people share what works for them, in any format they can.
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A milliard. The word is still used by countries that use the 'long scale', which has the nice property that a billion = (1 million)^2 and a quadrillion = (1 million)^4 etc. If you can count in greek this also means that an n-illion times an m-illion equals an (n+m)-illion.
This strikes me as a vastly superior way of doing things.. Any insight as to why things went 'the wrong way'?
[1] https://en.wikipedia.org/wiki/Names_of_large_numbers#Extensi...
MIT recorded a set of Calculus video courses back in 1970s that they have since made publicly available. It is taught by a lecturer named Herbert Gross. His style of lecturing is clear, he states why things are defined the way they are and derives everything from first principles. There is an unusual mix of rigor and focus on building understanding - where everything comes from. It also taught me that math is about r…
MIT recorded a set of Calculus video courses back in 1970s that they have since made publicly available. It is taught by a lecturer named Herbert Gross. His style of lecturing is clear, he states why things are defined the way they are and derives everything from first principles. There is an unusual mix of rigor and focus on building understanding - where everything comes from. It also taught me that math is about r…
at the opposite extreme, we should not always rely on logic and rigor when doing math - https://terrytao.wordpress.com/career-advice/there%E2%80%99s...
A formal system's territory is connected together by rigor; but whence comes the formal system itself? That has to be imagined.
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How did that happen? The required math courses got in the way?
Also attempting to finish CS BS but I have hit a wall in my late 30s unable to pass pre-calc. I shudder at the daunting levels of Calc that come after to the point that I'm debating switching majors just to "get a degree".