There is such a lot of wonderful maths in these presentations that I can't help feeling the need to whinge about a couple of things that seem out of place.
It's not always clear that you're talking about maths education as opposed to mathematics itself. Saying there is a "serious difficulty with modern mathematics" does make it sound like there is some flaw in modern mathematics itself and not just the way it's taught - which I'm sure is not what you meant to imply...
There seems to be quite a lot of talking about difficulties without saying exactly what they are. The fact that G H Hardy himself said there was a logical gap in most treatments of calculus, to do with trig functions, is very interesting - but only if you actually say what this logical gap in most treatments of calculus actually is!
Lastly if you're criticizing definitions, it seems important to make sure the objects your using are at least equally well defined.
I will be saying a lot more about the difficulties with the modern treatment (or lack of such) of circular functions when I eventually get around to discussing calculus and analysis. There is much more to be said in support of Hardy, but at this point the mathematics that I have built up is not sufficiently advanced to say these things clearly.
I wanted to gather in one place some of the main difficulties with angles, even if some of my comments involved some more advanced notions.
I need to clarify: modern mathematics itself is flawed, and many difficulties with mathematics education flow on from the logical weaknesses and confusions present in university and research level mathematics.
Academic mathematicians, like myself, have not done a proper job in ensuring that the basics of the subject are clear, well laid out and logical. We have, unfortunately, been bamboozled.
One of the purposes of this series is to set things on a proper course.
Oh wow - that's a whole other level, thanks. I'll wait with interest for the future videos, but I'm sure you can sympathise with non-research level maths students (like me) that the more extraordinary a claim, the clearer and more detailed the explanation will have to be to make us realize its truth...
Hope my nitpicking over things I don't understand is ok here - they're out of genuine curiosity. Just say if this is not the place - otherwise, I look forward to trying to understand...
There is such a lot of wonderful maths in these presentations that I can't help feeling the need to whinge about a couple of things that seem out of place.
It's not always clear that you're talking about maths education as opposed to mathematics itself. Saying there is a "serious difficulty with modern mathematics" does make it sound like there is some flaw in modern mathematics itself and not just the way it's taught - which I'm sure is not what you meant to imply...
mediteight 2 years ago
There seems to be quite a lot of talking about difficulties without saying exactly what they are. The fact that G H Hardy himself said there was a logical gap in most treatments of calculus, to do with trig functions, is very interesting - but only if you actually say what this logical gap in most treatments of calculus actually is!
Lastly if you're criticizing definitions, it seems important to make sure the objects your using are at least equally well defined.
Otherwise - eye opening!
mediteight 2 years ago
I will be saying a lot more about the difficulties with the modern treatment (or lack of such) of circular functions when I eventually get around to discussing calculus and analysis. There is much more to be said in support of Hardy, but at this point the mathematics that I have built up is not sufficiently advanced to say these things clearly.
I wanted to gather in one place some of the main difficulties with angles, even if some of my comments involved some more advanced notions.
njwildberger 2 years ago
Thanks Mediteight--
I need to clarify: modern mathematics itself is flawed, and many difficulties with mathematics education flow on from the logical weaknesses and confusions present in university and research level mathematics.
Academic mathematicians, like myself, have not done a proper job in ensuring that the basics of the subject are clear, well laid out and logical. We have, unfortunately, been bamboozled.
One of the purposes of this series is to set things on a proper course.
njwildberger 2 years ago
Oh wow - that's a whole other level, thanks. I'll wait with interest for the future videos, but I'm sure you can sympathise with non-research level maths students (like me) that the more extraordinary a claim, the clearer and more detailed the explanation will have to be to make us realize its truth...
Hope my nitpicking over things I don't understand is ok here - they're out of genuine curiosity. Just say if this is not the place - otherwise, I look forward to trying to understand...
mediteight 2 years ago