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Most math textbooks focus too much on proofs, as does this book. Data science is a broad field, but for most use cases formal proofs are not really needed. I wo

Most math textbooks focus too much on proofs, as does this book. Data science is a broad field, but for most use cases formal proofs are not really needed. I would prefer a text that focuses on examples and how/when to apply the theorems, rather than here is a theorem and here is a proof. Oh and here is another theorem and here is another proof. That isn't to say proofs are not important, but the focus of data science is usually on applied statistics.

catnaroek 10y ago
If you understand a mathematical theory, then you should be able to figure out on your own when that theory is applicable and how to apply it. So I can't help but read your comment as “I don't want to have to learn”.
- throw_away_777 10y ago
  You are being condescending and over simplifying the issues, try to not rush to judgment. I have spent a lot of work and time learning how to apply machine learning, so your comment was offensive and wrong. Applying mathematical theory is not just a matter of understanding the theory - in the real world there are a whole lot of reasons why you won't be able to apply the theory, even if you understand it perfectly. You will always have to make assumptions and approximations. I like learning, but I prefer learning for a purpose. Proofs don't really give you a practical understanding of the theory.
  - catnaroek 10y ago
    > You will always have to make assumptions and approximations. The ability to determine when an approximation is good enough comes from a solid understanding of the underlying theory. For example, we understand that special relativity doesn't invalidate classical mechanics, because, in the limit, as speeds and energies become arbitrarily small, their predictions coincide. But, if all you have is a bunch of computational recipes, even the tiniest unexpected thing renders you unable to solve problems. Anyway, I'm not saying that examples of applications are a bad thing. But proofs are indispensable. Even for “intuitive” people, proofs are necessary confirmation that their calculations will match their intuitions.
    - throw_away_777 10y ago
      You bring up an interesting example in special relativity. I am a physicist, and there isn't really a "proof" in the mathematical sense of special relativity. Googling it provides this quora answer: https://www.quora.com/What-is-the-proof-for-Special-Relativity https://www.quora.com/What-is-the-proof-for-Special-Relativi... There is no proof of special relativity, we believe it because it makes experimentally verified predictions. Your example of limiting cases of special relativity is how I wish statistics texts were taught - it isn't based on the "proof" of special relativity. Of course I know when special relativity is applicable and when it isn't (when beta = v / c is close to one, special relativistic effects are important, and gamma = 1 / sqrt(1 - beta*beta) indicates how good the approximation is). But I do agree that proofs are useful and needed, I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant. Of course this is just my personal preference.
      - catnaroek 10y ago
        > There is no proof of special relativity, we believe it because it makes experimentally verified predictions. Right, that's because special relativity is a scientific theory. A scientific theory has two components: a mathematical theory (in which you can perform a priori physically meaningless calculations) and a physical interpretation (which turns the results of such calculations into predictions). However, statistics isn't a scientific theory. It's just math, and like all math, it's about itself and nothing else. It doesn't make sense to “experimentally confirm a mathematical theory”, because, without a physical interpretation, math doesn't make any predictions about the real world. > I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant. Then you're looking for books on applications. That's fine. But a book whose subject matter is a mathematical theory (it even has “foundations” in the title!) can't relegate proofs to appendices.
yummyfajitas 10y ago
I would prefer a text that focuses on the java standard library and shows how/when to apply a java.util.List<T>. That's far better than a Data Structures and Algorithms book that teaches you how you'd implement a linked list or other complex data structure. 90% of the time I use the java standard lib. Sometimes I need something almost the same, but a bit different. The difference between a middling developer and a good one is what happens in that 10% of the time. The same applies to proofs. What happens when you need something that's almost like the theorem, but not exactly? Can you tweak the proof to apply to your case, or alternately recognize that it can't be fixed and you need to do something different?
- mindcrime 10y ago
  I don't think anybody is saying that proofs don't matter, or that they aren't important. But it looks like you're choosing to focus on uncommon edge cases. I think a lot of people would question that line of thinking. It's almost like the old saw about how you don't need a degree in mechanical engineering in order to effectively use a car. But if you need something "almost like a car, but also kinda like a bulldozer" then you're out of luck. OK, how often does that scenario come up? Enough to care about? For most people, the answer is a resounding "no". Likewise, one can certainly apply a lot of mathematical techniques in a lot of situations, without proving new theorems. So there are edge cases... not the end of the world. Deal with it when it comes up.
- throw_away_777 10y ago
  I agree that knowing and understanding the fundamental math is important, but I think proof based texts are not a good way to learn math. A lot of people get turned off from math because of how it is taught. Personally I learn best by doing problems and seeing examples. Proofs are useful, but I want to know why the theorem is important before I learn how to prove it is true.
mindcrime 10y ago
Most math textbooks focus too much on proofs, as does this book. My problem with (most) maths texts isn't the proofs in and of themselves, it's just that they make so many assumptions about what you already know, and don't always state those assumptions clearly. And then too many maths texts, to me, fail to include enough expository text to explain the math and provide more of an intuition around what's going on. That said, I also agree with you that more examples in terms of applications would be nice. Learning really abstract stuff in isolation is OK, but it's always nice to have a few (or more than a few) examples of how to apply the technique to something concrete.
- sn9 10y ago
  Most well-written textbooks list the assumed background knowledge in the preface/introduction. Just skimming that part for the word "prerequisites" or "background" will usually find it.
  - mindcrime 10y ago
    Most well-written textbooks list the assumed background knowledge in the preface/introduction. Exactly. And a surprisingly large number of books aren't well-written, exactly because they don't include that bit.
    - sn9 10y ago
      Over 90% of any medium is garbage which is why you have to take the time to make sure you're spending the time on the best material. For most well-studied topics, finding the books that are good enough is not that difficult.
  - eli_gottlieb 10y ago
    > Most well-written textbooks list the assumed background knowledge in the preface/introduction. Most actually-existing books simply write that the student should be "mathematically mature", by which they mean that the student should know all of multivariable and vector calculus, most of differential equations, much of optimization, and some amount of analysis. And also be able to do proofs, and also have good intuition. In short, most "applied" math textbooks aim themselves at Master's degree students or beginning PhD students in math itself, to teach material which can actually be taught to anyone with a bachelor's in science or engineering (except possibly computer scientists, whose continuous maths degrade because we never use them). Ironically, the first math textbook I've ever picked up that didn't assume far too much background was my real analysis textbook, because analysis teachers assume that they are teaching the gateway course to "real math" and have to educate little babbies who only just got done with calculus.

catnaroek 10y ago

If you understand a mathematical theory, then you should be able to figure out on your own when that theory is applicable and how to apply it. So I can't help but read your comment as “I don't want to have to learn”.

throw_away_777 10y ago
You are being condescending and over simplifying the issues, try to not rush to judgment. I have spent a lot of work and time learning how to apply machine learning, so your comment was offensive and wrong. Applying mathematical theory is not just a matter of understanding the theory - in the real world there are a whole lot of reasons why you won't be able to apply the theory, even if you understand it perfectly. You will always have to make assumptions and approximations. I like learning, but I prefer learning for a purpose. Proofs don't really give you a practical understanding of the theory.
- catnaroek 10y ago
  > You will always have to make assumptions and approximations. The ability to determine when an approximation is good enough comes from a solid understanding of the underlying theory. For example, we understand that special relativity doesn't invalidate classical mechanics, because, in the limit, as speeds and energies become arbitrarily small, their predictions coincide. But, if all you have is a bunch of computational recipes, even the tiniest unexpected thing renders you unable to solve problems. Anyway, I'm not saying that examples of applications are a bad thing. But proofs are indispensable. Even for “intuitive” people, proofs are necessary confirmation that their calculations will match their intuitions.
  - throw_away_777 10y ago
    You bring up an interesting example in special relativity. I am a physicist, and there isn't really a "proof" in the mathematical sense of special relativity. Googling it provides this quora answer: https://www.quora.com/What-is-the-proof-for-Special-Relativity https://www.quora.com/What-is-the-proof-for-Special-Relativi... There is no proof of special relativity, we believe it because it makes experimentally verified predictions. Your example of limiting cases of special relativity is how I wish statistics texts were taught - it isn't based on the "proof" of special relativity. Of course I know when special relativity is applicable and when it isn't (when beta = v / c is close to one, special relativistic effects are important, and gamma = 1 / sqrt(1 - beta*beta) indicates how good the approximation is). But I do agree that proofs are useful and needed, I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant. Of course this is just my personal preference.
    - catnaroek 10y ago
      > There is no proof of special relativity, we believe it because it makes experimentally verified predictions. Right, that's because special relativity is a scientific theory. A scientific theory has two components: a mathematical theory (in which you can perform a priori physically meaningless calculations) and a physical interpretation (which turns the results of such calculations into predictions). However, statistics isn't a scientific theory. It's just math, and like all math, it's about itself and nothing else. It doesn't make sense to “experimentally confirm a mathematical theory”, because, without a physical interpretation, math doesn't make any predictions about the real world. > I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant. Then you're looking for books on applications. That's fine. But a book whose subject matter is a mathematical theory (it even has “foundations” in the title!) can't relegate proofs to appendices.

yummyfajitas 10y ago

I would prefer a text that focuses on the java standard library and shows how/when to apply a java.util.List<T>. That's far better than a Data Structures and Algorithms book that teaches you how you'd implement a linked list or other complex data structure. 90% of the time I use the java standard lib. Sometimes I need something almost the same, but a bit different. The difference between a middling developer and a good one is what happens in that 10% of the time. The same applies to proofs. What happens when you need something that's almost like the theorem, but not exactly? Can you tweak the proof to apply to your case, or alternately recognize that it can't be fixed and you need to do something different?

mindcrime 10y ago
I don't think anybody is saying that proofs don't matter, or that they aren't important. But it looks like you're choosing to focus on uncommon edge cases. I think a lot of people would question that line of thinking. It's almost like the old saw about how you don't need a degree in mechanical engineering in order to effectively use a car. But if you need something "almost like a car, but also kinda like a bulldozer" then you're out of luck. OK, how often does that scenario come up? Enough to care about? For most people, the answer is a resounding "no". Likewise, one can certainly apply a lot of mathematical techniques in a lot of situations, without proving new theorems. So there are edge cases... not the end of the world. Deal with it when it comes up.
throw_away_777 10y ago
I agree that knowing and understanding the fundamental math is important, but I think proof based texts are not a good way to learn math. A lot of people get turned off from math because of how it is taught. Personally I learn best by doing problems and seeing examples. Proofs are useful, but I want to know why the theorem is important before I learn how to prove it is true.

mindcrime 10y ago

Most math textbooks focus too much on proofs, as does this book. My problem with (most) maths texts isn't the proofs in and of themselves, it's just that they make so many assumptions about what you already know, and don't always state those assumptions clearly. And then too many maths texts, to me, fail to include enough expository text to explain the math and provide more of an intuition around what's going on. That said, I also agree with you that more examples in terms of applications would be nice. Learning really abstract stuff in isolation is OK, but it's always nice to have a few (or more than a few) examples of how to apply the technique to something concrete.

sn9 10y ago
Most well-written textbooks list the assumed background knowledge in the preface/introduction. Just skimming that part for the word "prerequisites" or "background" will usually find it.
- mindcrime 10y ago
  Most well-written textbooks list the assumed background knowledge in the preface/introduction. Exactly. And a surprisingly large number of books aren't well-written, exactly because they don't include that bit.
  - sn9 10y ago
    Over 90% of any medium is garbage which is why you have to take the time to make sure you're spending the time on the best material. For most well-studied topics, finding the books that are good enough is not that difficult.
- eli_gottlieb 10y ago
  > Most well-written textbooks list the assumed background knowledge in the preface/introduction. Most actually-existing books simply write that the student should be "mathematically mature", by which they mean that the student should know all of multivariable and vector calculus, most of differential equations, much of optimization, and some amount of analysis. And also be able to do proofs, and also have good intuition. In short, most "applied" math textbooks aim themselves at Master's degree students or beginning PhD students in math itself, to teach material which can actually be taught to anyone with a bachelor's in science or engineering (except possibly computer scientists, whose continuous maths degrade because we never use them). Ironically, the first math textbook I've ever picked up that didn't assume far too much background was my real analysis textbook, because analysis teachers assume that they are teaching the gateway course to "real math" and have to educate little babbies who only just got done with calculus.