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On the math page that he linked he has some ideas: > 1) When you get older math got harder. > 2) When math got more abstract it got harder. Blame Grothendieck

On the math page that he linked he has some ideas: > 1) When you get older math got harder. > 2) When math got more abstract it got harder. Blame Grothendieck. > 3) When math stopped being tied to the real work it got harder. Blame Hardy. > 4) Math has always been hard. We NOW understand some of the older math better so it seems easy to us, but it wasn't at the time. > 5) With the web and more people working in math, new results come out faster so its harder to keep up. > 6) All fields of math have a period of time when they are easy, at the beginning, and then as the low-hanging fruit gets picked it gets harder and harder. So if a NEW branch was started it might initially be easy. Counterthought- even a new branch might be hard now since it can draw on so much prior math. Also, the low hanging fruit may be picked rather quickly. Personally I think it is mostly 2 and 3. And a lot of it has to do with how math is communicated. The very abstract way that math textbooks use to communicate is not how most mathematicians think when solving problems. A lot of things I struggled to learn from books became much more understandable once someone gave me the little insight I needed to make sense of it in conversation.

mcguire 5y ago
I don't think 6 has applied to math for hundreds of years, but I do see it as having applied to CS relatively recently.
bumby 5y ago
>A lot of things I struggled to learn from books became much more understandable once someone gave me the little insight I needed to make sense of it in conversation. Don't just leave us hanging :-) Care to elaborate on what helped gain that insight?
ericbarrett 5y ago
I completely agree regarding books. I've been refreshing my university undergrad-level math, and older textbooks have been much more accessible than most written after (roughly) the 1990s, even though the subjects haven't changed at all—this isn't avant-garde stuff. Seems no field is exempt from the modern "churn for churn's sake" phenomenon.
- bigthymer 5y ago
  > older textbooks have been much more accessible than most written after (roughly) the 1990s Why? How are they different?
- hungryforcodes 5y ago
  In Grade 12, I had a revelation when a good friend explained to me in 5 minutes that trigonometry was largely just ratios. I guess either it hadn't been explained to me before properly or I didn't understand it at a DEEP level, but when he laid it all out for me on the board, it was suddenly all so simple. I had three reactions: 1) I was elated to finally understand sin and cosin. 2) I was thankful for him and also amazed at how simple it was in 5 minutes to explain something that seemed to have eluded me for 5 years of highschool. It made me wonder alot what other difficult subjects could be easily taught by just a good explanation. 3) I wondered why math class had failed so specifically to give me that knowledge, or at least test me to see if I understood it and if I didn't to ensure that I did understand it. The answer I think was math class just encouraged me to use a calculator, thus short circuiting any real understanding. Anyways he went on to be a physics professor at Cambridge. I always appreciated that moment.
  - dataflow 5y ago
    How were sine and cosine taught to you as anything other than ratios? I just assumed they were always taught as "opposite over hypotenuse" and "adjacent over hypotenuse" etc.
    - hungryforcodes 5y ago
      They were clear to me intellectually, but the way he explained it made me SEE the ratios. Instead of just being abstract functions, there was a clear relationship between the angles in the triangle. After that I was able to reason about them, where as before they were just things I plugged into my calculator.
      - mixmastamyk 5y ago
Jensson 5y ago
How much math have you studied? I think they are talking about grad level math courses where things starts to get really abstract. Calculus, linear algebra and discrete math are all super concrete, pretty sure they are talking about things described as "generalized abstract nonsense": https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense Try reading the example on that page. That is what makes math hard. Not the way calculus is described.
- jeffreyrogers 5y ago
  I had a math minor in college and took some upper level analysis courses, so I know what abstract means in the context of higher mathematics. I wasn't talking about calculus.
  - Jensson 5y ago
    But how would you study abstract math without those abstractions? The math is just abstractions and nothing else, the abstractions are what you study, there is no concrete thing there to describe. So your original comment tells me you haven't studied a lot of very abstract math since you still want those explanations, but they stop existing after a point. > took some upper level analysis courses Real Analysis is still not very abstract, but yeah that is a step. The next step is to remove everything concrete about it and just study abstractions. And then you start studying abstractions of those abstractions, hence "abstract nonsense". Maybe sometimes in the future someone will invent abstractions of those. Edit: Meant the stuff you learn in Rudins principles of mathematical analysis. Not sure exactly what the course would be called in an English class, but I called it real analysis here.
    - jeffreyrogers 5y ago
      > But how would you study abstract math without those abstractions? I'm not saying you would, I'm just saying the abstractions are directly related to why its harder. > since you still want those explanations, but they stop existing after a point. You're misunderstanding me. > The next step is to remove everything concrete about it and just study abstractions. And then you start studying abstractions of those abstractions, hence "abstract nonsense". Maybe sometimes in the future someone will invent abstractions of those. Yes, I understand all this. I have the textbooks and I've read parts of them. The reason I didn't study math more seriously (I considered it) is because I realized I didn't have the brain for it and am not that interested in studying really abstract stuff.
hinkley 5y ago
Randall Munroe has a strip about how all the math most people need is how to fairly split a bill for a birthday dinner (so that the guest of honor doesn't pay). I sometimes wonder if we'd be better off telling teenagers that the motivation they need to learn math is self-preservation. There are always people who are going to try to trick you with bad math (from misleading cost/benefit analysis, to financial malfeasance, to bad faith public policy debates) and if you know some math they can't do that to you. The "you give Timmy five apples" shtick is almost anathema to that. You've practically built a narrative for the kid to get distracted from the point of the exercise. I don't like Timmy, or I don't know who the hell Timmy is. I'm not giving him five of my apples. Timmy can go fuck himself. Get your own damn apples. Better: you put twenty skittles into a cup. You drop the cup, and there are only 5 skittles left in the cup. How many are on the floor/how many do you need to replace the lost ones?
- emodendroket 5y ago
  That would seem to imply that there's little point in learning algebra, let alone anything more advanced than that.
  - smegger001 5y ago
    well to be fair i have never had to find the area under a curve, and have had to use the quadratic formula like once. and never had to find the log^10 of anything I use arithmetic logic and geometry and occasionally some basic statistics.
  - zamadatix 5y ago
    For most people there is little point in doing so from a "I really benefitted from knowing how to use this actual math" perspective. Fantastic for expanding reasoning and logical thinking capabilities, being able to come up with ways to reason about previously unseen logical problems from experience on learning how to solve many others previously. Not because finding a particular angle in an irregular nonagon from some other measures will be useful math to know itself.
  - quinnjh 5y ago

mcguire 5y ago

I don't think 6 has applied to math for hundreds of years, but I do see it as having applied to CS relatively recently.

bumby 5y ago

>A lot of things I struggled to learn from books became much more understandable once someone gave me the little insight I needed to make sense of it in conversation. Don't just leave us hanging :-) Care to elaborate on what helped gain that insight?

ericbarrett 5y ago

I completely agree regarding books. I've been refreshing my university undergrad-level math, and older textbooks have been much more accessible than most written after (roughly) the 1990s, even though the subjects haven't changed at all—this isn't avant-garde stuff. Seems no field is exempt from the modern "churn for churn's sake" phenomenon.

bigthymer 5y ago
> older textbooks have been much more accessible than most written after (roughly) the 1990s Why? How are they different?
hungryforcodes 5y ago
In Grade 12, I had a revelation when a good friend explained to me in 5 minutes that trigonometry was largely just ratios. I guess either it hadn't been explained to me before properly or I didn't understand it at a DEEP level, but when he laid it all out for me on the board, it was suddenly all so simple. I had three reactions: 1) I was elated to finally understand sin and cosin. 2) I was thankful for him and also amazed at how simple it was in 5 minutes to explain something that seemed to have eluded me for 5 years of highschool. It made me wonder alot what other difficult subjects could be easily taught by just a good explanation. 3) I wondered why math class had failed so specifically to give me that knowledge, or at least test me to see if I understood it and if I didn't to ensure that I did understand it. The answer I think was math class just encouraged me to use a calculator, thus short circuiting any real understanding. Anyways he went on to be a physics professor at Cambridge. I always appreciated that moment.
- dataflow 5y ago
  How were sine and cosine taught to you as anything other than ratios? I just assumed they were always taught as "opposite over hypotenuse" and "adjacent over hypotenuse" etc.
  - hungryforcodes 5y ago
    They were clear to me intellectually, but the way he explained it made me SEE the ratios. Instead of just being abstract functions, there was a clear relationship between the angles in the triangle. After that I was able to reason about them, where as before they were just things I plugged into my calculator.
    - mixmastamyk 5y ago

Jensson 5y ago

How much math have you studied? I think they are talking about grad level math courses where things starts to get really abstract. Calculus, linear algebra and discrete math are all super concrete, pretty sure they are talking about things described as "generalized abstract nonsense": https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense Try reading the example on that page. That is what makes math hard. Not the way calculus is described.

jeffreyrogers 5y ago
I had a math minor in college and took some upper level analysis courses, so I know what abstract means in the context of higher mathematics. I wasn't talking about calculus.
- Jensson 5y ago
  But how would you study abstract math without those abstractions? The math is just abstractions and nothing else, the abstractions are what you study, there is no concrete thing there to describe. So your original comment tells me you haven't studied a lot of very abstract math since you still want those explanations, but they stop existing after a point. > took some upper level analysis courses Real Analysis is still not very abstract, but yeah that is a step. The next step is to remove everything concrete about it and just study abstractions. And then you start studying abstractions of those abstractions, hence "abstract nonsense". Maybe sometimes in the future someone will invent abstractions of those. Edit: Meant the stuff you learn in Rudins principles of mathematical analysis. Not sure exactly what the course would be called in an English class, but I called it real analysis here.
  - jeffreyrogers 5y ago
    > But how would you study abstract math without those abstractions? I'm not saying you would, I'm just saying the abstractions are directly related to why its harder. > since you still want those explanations, but they stop existing after a point. You're misunderstanding me. > The next step is to remove everything concrete about it and just study abstractions. And then you start studying abstractions of those abstractions, hence "abstract nonsense". Maybe sometimes in the future someone will invent abstractions of those. Yes, I understand all this. I have the textbooks and I've read parts of them. The reason I didn't study math more seriously (I considered it) is because I realized I didn't have the brain for it and am not that interested in studying really abstract stuff.

hinkley 5y ago

Randall Munroe has a strip about how all the math most people need is how to fairly split a bill for a birthday dinner (so that the guest of honor doesn't pay). I sometimes wonder if we'd be better off telling teenagers that the motivation they need to learn math is self-preservation. There are always people who are going to try to trick you with bad math (from misleading cost/benefit analysis, to financial malfeasance, to bad faith public policy debates) and if you know some math they can't do that to you. The "you give Timmy five apples" shtick is almost anathema to that. You've practically built a narrative for the kid to get distracted from the point of the exercise. I don't like Timmy, or I don't know who the hell Timmy is. I'm not giving him five of my apples. Timmy can go fuck himself. Get your own damn apples. Better: you put twenty skittles into a cup. You drop the cup, and there are only 5 skittles left in the cup. How many are on the floor/how many do you need to replace the lost ones?

emodendroket 5y ago
That would seem to imply that there's little point in learning algebra, let alone anything more advanced than that.
- smegger001 5y ago
  well to be fair i have never had to find the area under a curve, and have had to use the quadratic formula like once. and never had to find the log^10 of anything I use arithmetic logic and geometry and occasionally some basic statistics.
- zamadatix 5y ago
  For most people there is little point in doing so from a "I really benefitted from knowing how to use this actual math" perspective. Fantastic for expanding reasoning and logical thinking capabilities, being able to come up with ways to reason about previously unseen logical problems from experience on learning how to solve many others previously. Not because finding a particular angle in an irregular nonagon from some other measures will be useful math to know itself.
- quinnjh 5y ago