On Tuesday, 13 April 2021 at 06:23:36 UTC-4, Quantum Bubbles wrote:
> Mr Gabriel's "New Calculus" will not give you an advantage in understanding mathematics and will stall your mathematical development if you take it too seriously. As someone with multiple mathematics degrees and a teaching qualification in mathematics, I am genuinely disturbed by the idea that students, perhaps suffering from anxiety, lack of self-confidence and other very common difficulties, might have their precious time and effort wasted by this.
>
> Calculus and analysis can be tricky subjects to learn, and even trickier to master. There is a lot of material to digest; many definitions, many techniques, the calculus books are sometimes 1000 pages long, and the analysis books seem to be packed with a blizzard of inequalities. You might have spent days learning an idea and then weeks later it feels as if you have forgotten it all.
>
> It can look very daunting. This is a perfectly normal feeling and neither signifies that you cannot do it, nor that the mathematics community does not know what it is doing. There are many books on the same subjects, and different students will be better served by different choices of books, or different ways of presenting the same ideas. Teachers cannot always cater for every preference at once; but that does not mean there isn't a good book FOR YOU out there.
>
> It is perfectly normal for it to take years to master these topics, and it is perfectly normal to struggle at various stages. It is normal to struggle at the beginning when a mathematical concept is new. It is normal to struggle when the topics have become more abstract. It is normal to struggle when connections between different ideas need to be explored, because the cognitive load is demanding. It is normal for memories to fade after the pressure of exams has ended, unless you practice to help solidify those memories afterwards.
>
> This is true for most people; it doesn't really matter which university you go to, or how smart you are (even genius's sometimes need feedback from other experts: Leonhard Euler and John Nash did for example).
>
> The solution to this is to find study methods that work FOR YOU (which might not be the same as what works for someone else), and which books work best FOR YOU. I will give what I hope is a helpful list for both at the end of this post, including some genuinely useful YouTube channels. Once you find material and approaches that work for you, you can be surprised at how much more smoothly things go; but it is never truly easy, because maths is abstract and often subtle, and abstraction is always difficult.
>
> And it is perfectly normal for "philosophical" questions about a subject to sometimes occur to students, which they might find difficult to verbalize, and might not feel their book is directly addressing.
>
> Under no circumstances is the right response to this, to seek refuge in silly ancient Greek metaphysics, amateur psychology, and fantasizing that 150 years worth of mathematics experts missed the real value of an ancient Greek geometry book that had been regularly read and understood for centuries; nor that they are clueless about set theory that has been thought about for 150 years. Philosophers and mathematicians of many cultures have been thinking about abstract ideas of number and space for millennia; Mr Gabriel's ideas are neither deep, nor special, nor generally effective.
>
> Qualified mathematicians and teachers, including myself, have read Mr Gabriel's New Calculus and are not impressed by it. It can work in a very small class of situations, just like other pre-limit methods which existed in the 16th century, but beyond that not really. It also struggles with basic and important ideas like points of inflection, which are important in pure mathematics, game theory, the calculus of variations and other areas.
>
> Some examples of this kind of historical development can be found here, in this lovely 25 minute 1980's BBC/Open University video starring mathematics historian Jeremy Gray:
>
> https://www.youtube.com/watch?v=ObPg3ki9GOI
>
> Needless to say, Mr Gabriel's approach cannot be used to develop a useful calculus of variations nor indeed many of the standard methods of modern mathematical physics.
>
> Mr Gabriel is also mistaken in believing that he has a special insight into the concept of number. Elementary conceptions of number were common in higher mathematics pre-20th century. Leonhard Euler gave a much smoother and fruitful offering of one in his lovely book:
>
> "Elements of Algebra"
>
> which has been read since the 18th century, and can still be purchased in various abridged forms today. Euler was a genius and clear explainer, but no one (especially not Euler himself) claims he was a genius because of his smooth presentation of a number concept.
>
> POSITIVE RECOMMENDATIONS FOR STUDENTS
>
> 1] Concerning philosophy
>
> Most philosophical difficulties with early set theory have been satisfactorily addressed, but the detailed explanations can be very hard and dwelling on them during undergraduate studies is often not advisable because it can take far too long, and is better approached when you have more experience..
>
> There is nothing wrong with having philosophical opinions that differ from the mainstream per se, provided they are not dogmatic and provided they possess clarity. They should be clearly thought out, should not inhibit your ability to do core mathematics (algebra, calculus etc.) and shouldn't descend into silly conspiracy theories about academia.
>
> If, for example, students feel drawn to a more algorithmic and less Cantorian approach to mathematics than is normally presented, then I would suggest they look up Professor Harold M. Edwards, who has championed an algorithmic approach to mathematics for many years. His webpage is here:
>
> https://math.nyu.edu/faculty/edwardsd/
>
> In particular his book on elementary number theory:
>
> Higher Arithmetic: an algorithmic introduction to number theory
>
> might be suitable for such students. I do not share Professor Edwards' philosophical opinions, but at least he is intellectually sober and does genuinely good mathematics within the framework he advocates. So if you dislike or struggle with Cantorianism, then try Edwards instead, don't waste your time with vague ancient Greek metaphysics and ultra-finitist nonsense.
>
>
> 2] On psychology and learning
>
> Psychologists have been studying the learning of mathematics for decades, and some genuinely useful advice for studying more effectively has come out of it. Look up:
>
> - spaced practice and retrieval practice
>
> - Frayer models/diagrams
>
> - the brilliant lecture: https://www.youtube.com/watch?v=IlU-zDU6aQ0&t
>
> - the benefits of cardio-vascular exercise for improving study focus (the neurologist John Ratey has presented good stuff on this kind of thing)
>
> - The psychologist Anders Ericsson, who researched the development of expertise.
>
> Don't fall into the trap of thinking that if you stare at a problem for a few minutes and can't do it, that this means you aren't able to be very good at the subject. Many mathematicians need to ponder things at length and return to the same thing in an iterative fashion over a period of time. Even genius's like Nobel prize winning mathematical physicist Roger Penrose do this, and if you want to see what slow, fallible pondering can produce, look up his book:
>
> The Road to Reality
>
> which provides beautiful illustrations of what many of the mathematical ideas you study can be used to do in physics. You won't find a better guide than Penrose on that sort of thing. A brilliant teacher and a true inspiration.
>
> The book: How to Solve it, by G. Polya, is a classic short guide for undergraduates in developing problem solving skills in mathematics. Far more worthy of your time than Mr Gabriel's output.
>
>
> 3] Books on analysis
>
> The best book on analysis FOR YOU, is the one that is easiest to learn from FOR YOU. There is no universally suitable book and no book is perfect. An excellent book for students who lack confidence is:
>
> Fundamentals of Mathematical Analysis 2nd ed, by Rod Haggarty.
>
> It goes at a gentle pace, has diagrams, historical asides, and solutions to exercises. Good for self-study. It's only significant drawback is that it doesn't construct the real numbers. But for that, you can look up explanations of Dedekind cuts on the internet.
>
> I found my first and second year courses on real analysis quite difficult at first, but I bought Haggarty's book, threw myself into it and I aced my courses and went on to study the Calculus of Variations at MSc level later on. I didn't disrupt my education by indulging the silly idea that real analysis was built on obvious delusions and lies. That way represents the slow decent into bitterness and madness.
>
> A book which is somewhat more difficult (but not impossibly hard), but which has a beautiful style of exposition and does look at Dedekind cuts, is the classic text:
>
> A Course in Pure Mathematics (centenary edition), by G.H. Hardy
>
> Having both texts so that you can switch between them is a perfectly good strategy. Don't be afraid to have different books on the same subject. Professional mathematicians often do.
>
>
> 4] Genuinely helpful YouTube channels
>
> One of the least toxic places on YouTube, hosted by a university lecturer, lots of short advice videos about studying mathematics, book reviews and much else:
>
> The Math Sorcerer: https://www.youtube.com/user/themathsorcerer
>
> For good lecture series on mathematics and other topics, such as linear algebra, visit MITOpenCourseWare
>
> https://www.youtube.com/channel/UCEBb1b_L6zDS3xTUrIALZOw
>
> For some very beautiful visual presentations of mathematical ideas, you would find it hard to do better than 3Blue1Brown:
>
> https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
>
> Best of luck with your mathematics studies, and don't allow yourselves to be done a disservice by silly conspiracy theories. Seek assistance from your college or university if you are having difficulty with a particular subject or study in general; it is perfectly normal, and can be dealt with by seeking out good advice from your tutors, library or study support departments. Not by wasting your time on delirious rantings on the internet.
>
> Kindest Regards

On Thursday, 11 May 2023 at 14:41:56 UTC-4, Eram semper recta wrote:
> On Tuesday, 13 April 2021 at 06:23:36 UTC-4, Quantum Bubbles wrote:
> > Mr Gabriel's "New Calculus" will not give you an advantage in understanding mathematics and will stall your mathematical development if you take it too seriously. As someone with multiple mathematics degrees and a teaching qualification in mathematics, I am genuinely disturbed by the idea that students, perhaps suffering from anxiety, lack of self-confidence and other very common difficulties, might have their precious time and effort wasted by this.
> >
> > Calculus and analysis can be tricky subjects to learn, and even trickier to master. There is a lot of material to digest; many definitions, many techniques, the calculus books are sometimes 1000 pages long, and the analysis books seem to be packed with a blizzard of inequalities. You might have spent days learning an idea and then weeks later it feels as if you have forgotten it all.
> >
> > It can look very daunting. This is a perfectly normal feeling and neither signifies that you cannot do it, nor that the mathematics community does not know what it is doing. There are many books on the same subjects, and different students will be better served by different choices of books, or different ways of presenting the same ideas. Teachers cannot always cater for every preference at once; but that does not mean there isn't a good book FOR YOU out there.
> >
> > It is perfectly normal for it to take years to master these topics, and it is perfectly normal to struggle at various stages. It is normal to struggle at the beginning when a mathematical concept is new. It is normal to struggle when the topics have become more abstract. It is normal to struggle when connections between different ideas need to be explored, because the cognitive load is demanding. It is normal for memories to fade after the pressure of exams has ended, unless you practice to help solidify those memories afterwards.
> >
> > This is true for most people; it doesn't really matter which university you go to, or how smart you are (even genius's sometimes need feedback from other experts: Leonhard Euler and John Nash did for example).
> >
> > The solution to this is to find study methods that work FOR YOU (which might not be the same as what works for someone else), and which books work best FOR YOU. I will give what I hope is a helpful list for both at the end of this post, including some genuinely useful YouTube channels. Once you find material and approaches that work for you, you can be surprised at how much more smoothly things go; but it is never truly easy, because maths is abstract and often subtle, and abstraction is always difficult.
> >
> > And it is perfectly normal for "philosophical" questions about a subject to sometimes occur to students, which they might find difficult to verbalize, and might not feel their book is directly addressing.
> >
> > Under no circumstances is the right response to this, to seek refuge in silly ancient Greek metaphysics, amateur psychology, and fantasizing that 150 years worth of mathematics experts missed the real value of an ancient Greek geometry book that had been regularly read and understood for centuries; nor that they are clueless about set theory that has been thought about for 150 years. Philosophers and mathematicians of many cultures have been thinking about abstract ideas of number and space for millennia; Mr Gabriel's ideas are neither deep, nor special, nor generally effective.
> >
> > Qualified mathematicians and teachers, including myself, have read Mr Gabriel's New Calculus and are not impressed by it. It can work in a very small class of situations, just like other pre-limit methods which existed in the 16th century, but beyond that not really. It also struggles with basic and important ideas like points of inflection, which are important in pure mathematics, game theory, the calculus of variations and other areas.
> >
> > Some examples of this kind of historical development can be found here, in this lovely 25 minute 1980's BBC/Open University video starring mathematics historian Jeremy Gray:
> >
> > https://www.youtube.com/watch?v=ObPg3ki9GOI
> >
> > Needless to say, Mr Gabriel's approach cannot be used to develop a useful calculus of variations nor indeed many of the standard methods of modern mathematical physics.
> >
> > Mr Gabriel is also mistaken in believing that he has a special insight into the concept of number. Elementary conceptions of number were common in higher mathematics pre-20th century. Leonhard Euler gave a much smoother and fruitful offering of one in his lovely book:
> >
> > "Elements of Algebra"
> >
> > which has been read since the 18th century, and can still be purchased in various abridged forms today. Euler was a genius and clear explainer, but no one (especially not Euler himself) claims he was a genius because of his smooth presentation of a number concept.
> >
> > POSITIVE RECOMMENDATIONS FOR STUDENTS
> >
> > 1] Concerning philosophy
> >
> > Most philosophical difficulties with early set theory have been satisfactorily addressed, but the detailed explanations can be very hard and dwelling on them during undergraduate studies is often not advisable because it can take far too long, and is better approached when you have more experience.
> >
> > There is nothing wrong with having philosophical opinions that differ from the mainstream per se, provided they are not dogmatic and provided they possess clarity. They should be clearly thought out, should not inhibit your ability to do core mathematics (algebra, calculus etc.) and shouldn't descend into silly conspiracy theories about academia.
> >
> > If, for example, students feel drawn to a more algorithmic and less Cantorian approach to mathematics than is normally presented, then I would suggest they look up Professor Harold M. Edwards, who has championed an algorithmic approach to mathematics for many years. His webpage is here:
> >
> > https://math.nyu.edu/faculty/edwardsd/
> >
> > In particular his book on elementary number theory:
> >
> > Higher Arithmetic: an algorithmic introduction to number theory
> >
> > might be suitable for such students. I do not share Professor Edwards' philosophical opinions, but at least he is intellectually sober and does genuinely good mathematics within the framework he advocates. So if you dislike or struggle with Cantorianism, then try Edwards instead, don't waste your time with vague ancient Greek metaphysics and ultra-finitist nonsense.
> >
> >
> > 2] On psychology and learning
> >
> > Psychologists have been studying the learning of mathematics for decades, and some genuinely useful advice for studying more effectively has come out of it. Look up:
> >
> > - spaced practice and retrieval practice
> >
> > - Frayer models/diagrams
> >
> > - the brilliant lecture: https://www.youtube.com/watch?v=IlU-zDU6aQ0&t
> >
> > - the benefits of cardio-vascular exercise for improving study focus (the neurologist John Ratey has presented good stuff on this kind of thing)
> >
> > - The psychologist Anders Ericsson, who researched the development of expertise.
> >
> > Don't fall into the trap of thinking that if you stare at a problem for a few minutes and can't do it, that this means you aren't able to be very good at the subject. Many mathematicians need to ponder things at length and return to the same thing in an iterative fashion over a period of time. Even genius's like Nobel prize winning mathematical physicist Roger Penrose do this, and if you want to see what slow, fallible pondering can produce, look up his book:
> >
> > The Road to Reality
> >
> > which provides beautiful illustrations of what many of the mathematical ideas you study can be used to do in physics. You won't find a better guide than Penrose on that sort of thing. A brilliant teacher and a true inspiration.
> >
> > The book: How to Solve it, by G. Polya, is a classic short guide for undergraduates in developing problem solving skills in mathematics. Far more worthy of your time than Mr Gabriel's output.
> >
> >
> > 3] Books on analysis
> >
> > The best book on analysis FOR YOU, is the one that is easiest to learn from FOR YOU. There is no universally suitable book and no book is perfect. An excellent book for students who lack confidence is:
> >
> > Fundamentals of Mathematical Analysis 2nd ed, by Rod Haggarty.
> >
> > It goes at a gentle pace, has diagrams, historical asides, and solutions to exercises. Good for self-study. It's only significant drawback is that it doesn't construct the real numbers. But for that, you can look up explanations of Dedekind cuts on the internet.
> >
> > I found my first and second year courses on real analysis quite difficult at first, but I bought Haggarty's book, threw myself into it and I aced my courses and went on to study the Calculus of Variations at MSc level later on. I didn't disrupt my education by indulging the silly idea that real analysis was built on obvious delusions and lies. That way represents the slow decent into bitterness and madness.
> >
> > A book which is somewhat more difficult (but not impossibly hard), but which has a beautiful style of exposition and does look at Dedekind cuts, is the classic text:
> >
> > A Course in Pure Mathematics (centenary edition), by G.H. Hardy
> >
> > Having both texts so that you can switch between them is a perfectly good strategy. Don't be afraid to have different books on the same subject. Professional mathematicians often do.
> >
> >
> > 4] Genuinely helpful YouTube channels
> >
> > One of the least toxic places on YouTube, hosted by a university lecturer, lots of short advice videos about studying mathematics, book reviews and much else:
> >
> > The Math Sorcerer: https://www.youtube.com/user/themathsorcerer
> >
> > For good lecture series on mathematics and other topics, such as linear algebra, visit MITOpenCourseWare
> >
> > https://www.youtube.com/channel/UCEBb1b_L6zDS3xTUrIALZOw
> >
> > For some very beautiful visual presentations of mathematical ideas, you would find it hard to do better than 3Blue1Brown:
> >
> > https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
> >
> > Best of luck with your mathematics studies, and don't allow yourselves to be done a disservice by silly conspiracy theories. Seek assistance from your college or university if you are having difficulty with a particular subject or study in general; it is perfectly normal, and can be dealt with by seeking out good advice from your tutors, library or study support departments. Not by wasting your time on delirious rantings on the internet.
> >
> > Kindest Regards
>
> This lying anonymous bastard continues to spew out his vomit.
>
> Read for yourself what I have to say and not what some dumb UK cunt spews out here on sci.math:
>
> https://independent.academia.edu/JohnGabriel30

On Thursday, 11 May 2023 at 14:44:20 UTC-4, Eram semper recta wrote:
> On Thursday, 11 May 2023 at 14:41:56 UTC-4, Eram semper recta wrote:
> > On Tuesday, 13 April 2021 at 06:23:36 UTC-4, Quantum Bubbles wrote:
> > > Mr Gabriel's "New Calculus" will not give you an advantage in understanding mathematics and will stall your mathematical development if you take it too seriously. As someone with multiple mathematics degrees and a teaching qualification in mathematics, I am genuinely disturbed by the idea that students, perhaps suffering from anxiety, lack of self-confidence and other very common difficulties, might have their precious time and effort wasted by this.
> > >
> > > Calculus and analysis can be tricky subjects to learn, and even trickier to master. There is a lot of material to digest; many definitions, many techniques, the calculus books are sometimes 1000 pages long, and the analysis books seem to be packed with a blizzard of inequalities. You might have spent days learning an idea and then weeks later it feels as if you have forgotten it all.
> > >
> > > It can look very daunting. This is a perfectly normal feeling and neither signifies that you cannot do it, nor that the mathematics community does not know what it is doing. There are many books on the same subjects, and different students will be better served by different choices of books, or different ways of presenting the same ideas. Teachers cannot always cater for every preference at once; but that does not mean there isn't a good book FOR YOU out there.
> > >
> > > It is perfectly normal for it to take years to master these topics, and it is perfectly normal to struggle at various stages. It is normal to struggle at the beginning when a mathematical concept is new. It is normal to struggle when the topics have become more abstract. It is normal to struggle when connections between different ideas need to be explored, because the cognitive load is demanding. It is normal for memories to fade after the pressure of exams has ended, unless you practice to help solidify those memories afterwards.
> > >
> > > This is true for most people; it doesn't really matter which university you go to, or how smart you are (even genius's sometimes need feedback from other experts: Leonhard Euler and John Nash did for example).
> > >
> > > The solution to this is to find study methods that work FOR YOU (which might not be the same as what works for someone else), and which books work best FOR YOU. I will give what I hope is a helpful list for both at the end of this post, including some genuinely useful YouTube channels. Once you find material and approaches that work for you, you can be surprised at how much more smoothly things go; but it is never truly easy, because maths is abstract and often subtle, and abstraction is always difficult.
> > >
> > > And it is perfectly normal for "philosophical" questions about a subject to sometimes occur to students, which they might find difficult to verbalize, and might not feel their book is directly addressing.
> > >
> > > Under no circumstances is the right response to this, to seek refuge in silly ancient Greek metaphysics, amateur psychology, and fantasizing that 150 years worth of mathematics experts missed the real value of an ancient Greek geometry book that had been regularly read and understood for centuries; nor that they are clueless about set theory that has been thought about for 150 years. Philosophers and mathematicians of many cultures have been thinking about abstract ideas of number and space for millennia; Mr Gabriel's ideas are neither deep, nor special, nor generally effective.
> > >
> > > Qualified mathematicians and teachers, including myself, have read Mr Gabriel's New Calculus and are not impressed by it. It can work in a very small class of situations, just like other pre-limit methods which existed in the 16th century, but beyond that not really. It also struggles with basic and important ideas like points of inflection, which are important in pure mathematics, game theory, the calculus of variations and other areas.
> > >
> > > Some examples of this kind of historical development can be found here, in this lovely 25 minute 1980's BBC/Open University video starring mathematics historian Jeremy Gray:
> > >
> > > https://www.youtube.com/watch?v=ObPg3ki9GOI
> > >
> > > Needless to say, Mr Gabriel's approach cannot be used to develop a useful calculus of variations nor indeed many of the standard methods of modern mathematical physics.
> > >
> > > Mr Gabriel is also mistaken in believing that he has a special insight into the concept of number. Elementary conceptions of number were common in higher mathematics pre-20th century. Leonhard Euler gave a much smoother and fruitful offering of one in his lovely book:
> > >
> > > "Elements of Algebra"
> > >
> > > which has been read since the 18th century, and can still be purchased in various abridged forms today. Euler was a genius and clear explainer, but no one (especially not Euler himself) claims he was a genius because of his smooth presentation of a number concept.
> > >
> > > POSITIVE RECOMMENDATIONS FOR STUDENTS
> > >
> > > 1] Concerning philosophy
> > >
> > > Most philosophical difficulties with early set theory have been satisfactorily addressed, but the detailed explanations can be very hard and dwelling on them during undergraduate studies is often not advisable because it can take far too long, and is better approached when you have more experience.
> > >
> > > There is nothing wrong with having philosophical opinions that differ from the mainstream per se, provided they are not dogmatic and provided they possess clarity. They should be clearly thought out, should not inhibit your ability to do core mathematics (algebra, calculus etc.) and shouldn't descend into silly conspiracy theories about academia.
> > >
> > > If, for example, students feel drawn to a more algorithmic and less Cantorian approach to mathematics than is normally presented, then I would suggest they look up Professor Harold M. Edwards, who has championed an algorithmic approach to mathematics for many years. His webpage is here:
> > >
> > > https://math.nyu.edu/faculty/edwardsd/
> > >
> > > In particular his book on elementary number theory:
> > >
> > > Higher Arithmetic: an algorithmic introduction to number theory
> > >
> > > might be suitable for such students. I do not share Professor Edwards' philosophical opinions, but at least he is intellectually sober and does genuinely good mathematics within the framework he advocates. So if you dislike or struggle with Cantorianism, then try Edwards instead, don't waste your time with vague ancient Greek metaphysics and ultra-finitist nonsense.
> > >
> > >
> > > 2] On psychology and learning
> > >
> > > Psychologists have been studying the learning of mathematics for decades, and some genuinely useful advice for studying more effectively has come out of it. Look up:
> > >
> > > - spaced practice and retrieval practice
> > >
> > > - Frayer models/diagrams
> > >
> > > - the brilliant lecture: https://www.youtube.com/watch?v=IlU-zDU6aQ0&t
> > >
> > > - the benefits of cardio-vascular exercise for improving study focus (the neurologist John Ratey has presented good stuff on this kind of thing)
> > >
> > > - The psychologist Anders Ericsson, who researched the development of expertise.
> > >
> > > Don't fall into the trap of thinking that if you stare at a problem for a few minutes and can't do it, that this means you aren't able to be very good at the subject. Many mathematicians need to ponder things at length and return to the same thing in an iterative fashion over a period of time. Even genius's like Nobel prize winning mathematical physicist Roger Penrose do this, and if you want to see what slow, fallible pondering can produce, look up his book:
> > >
> > > The Road to Reality
> > >
> > > which provides beautiful illustrations of what many of the mathematical ideas you study can be used to do in physics. You won't find a better guide than Penrose on that sort of thing. A brilliant teacher and a true inspiration.
> > >
> > > The book: How to Solve it, by G. Polya, is a classic short guide for undergraduates in developing problem solving skills in mathematics. Far more worthy of your time than Mr Gabriel's output.
> > >
> > >
> > > 3] Books on analysis
> > >
> > > The best book on analysis FOR YOU, is the one that is easiest to learn from FOR YOU. There is no universally suitable book and no book is perfect. An excellent book for students who lack confidence is:
> > >
> > > Fundamentals of Mathematical Analysis 2nd ed, by Rod Haggarty.
> > >
> > > It goes at a gentle pace, has diagrams, historical asides, and solutions to exercises. Good for self-study. It's only significant drawback is that it doesn't construct the real numbers. But for that, you can look up explanations of Dedekind cuts on the internet.
> > >
> > > I found my first and second year courses on real analysis quite difficult at first, but I bought Haggarty's book, threw myself into it and I aced my courses and went on to study the Calculus of Variations at MSc level later on. I didn't disrupt my education by indulging the silly idea that real analysis was built on obvious delusions and lies. That way represents the slow decent into bitterness and madness.
> > >
> > > A book which is somewhat more difficult (but not impossibly hard), but which has a beautiful style of exposition and does look at Dedekind cuts, is the classic text:
> > >
> > > A Course in Pure Mathematics (centenary edition), by G.H. Hardy
> > >
> > > Having both texts so that you can switch between them is a perfectly good strategy. Don't be afraid to have different books on the same subject. Professional mathematicians often do.
> > >
> > >
> > > 4] Genuinely helpful YouTube channels
> > >
> > > One of the least toxic places on YouTube, hosted by a university lecturer, lots of short advice videos about studying mathematics, book reviews and much else:
> > >
> > > The Math Sorcerer: https://www.youtube.com/user/themathsorcerer
> > >
> > > For good lecture series on mathematics and other topics, such as linear algebra, visit MITOpenCourseWare
> > >
> > > https://www.youtube.com/channel/UCEBb1b_L6zDS3xTUrIALZOw
> > >
> > > For some very beautiful visual presentations of mathematical ideas, you would find it hard to do better than 3Blue1Brown:
> > >
> > > https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
> > >
> > > Best of luck with your mathematics studies, and don't allow yourselves to be done a disservice by silly conspiracy theories. Seek assistance from your college or university if you are having difficulty with a particular subject or study in general; it is perfectly normal, and can be dealt with by seeking out good advice from your tutors, library or study support departments. Not by wasting your time on delirious rantings on the internet.
> > >
> > > Kindest Regards
> >
> > This lying anonymous bastard continues to spew out his vomit.
> >
> > Read for yourself what I have to say and not what some dumb UK cunt spews out here on sci.math:
> >
> > https://independent.academia.edu/JohnGabriel30
> Begin with my historic geometric theorem of January 2020 which exposes the lie that mainstream calculus was ever rigorous:
>
> https://www.academia.edu/62358358/My_historic_geometric_theorem_of_January_2020
>
> https://www.academia.edu/79881709/Six_simple_reasons_why_the_mainstream_derivative_definition_of_calculus_is_flawed
>
> and much more at the above link: https://independent.academia.edu/JohnGabriel30

On Monday, 26 June 2023 at 11:39:27 UTC-4, Eram semper recta wrote:
> On Thursday, 11 May 2023 at 14:44:20 UTC-4, Eram semper recta wrote:
> > On Thursday, 11 May 2023 at 14:41:56 UTC-4, Eram semper recta wrote:
> > > On Tuesday, 13 April 2021 at 06:23:36 UTC-4, Quantum Bubbles wrote:
> > > > Mr Gabriel's "New Calculus" will not give you an advantage in understanding mathematics and will stall your mathematical development if you take it too seriously. As someone with multiple mathematics degrees and a teaching qualification in mathematics, I am genuinely disturbed by the idea that students, perhaps suffering from anxiety, lack of self-confidence and other very common difficulties, might have their precious time and effort wasted by this.
> > > >
> > > > Calculus and analysis can be tricky subjects to learn, and even trickier to master. There is a lot of material to digest; many definitions, many techniques, the calculus books are sometimes 1000 pages long, and the analysis books seem to be packed with a blizzard of inequalities. You might have spent days learning an idea and then weeks later it feels as if you have forgotten it all.
> > > >
> > > > It can look very daunting. This is a perfectly normal feeling and neither signifies that you cannot do it, nor that the mathematics community does not know what it is doing. There are many books on the same subjects, and different students will be better served by different choices of books, or different ways of presenting the same ideas. Teachers cannot always cater for every preference at once; but that does not mean there isn't a good book FOR YOU out there.
> > > >
> > > > It is perfectly normal for it to take years to master these topics, and it is perfectly normal to struggle at various stages. It is normal to struggle at the beginning when a mathematical concept is new. It is normal to struggle when the topics have become more abstract. It is normal to struggle when connections between different ideas need to be explored, because the cognitive load is demanding. It is normal for memories to fade after the pressure of exams has ended, unless you practice to help solidify those memories afterwards.
> > > >
> > > > This is true for most people; it doesn't really matter which university you go to, or how smart you are (even genius's sometimes need feedback from other experts: Leonhard Euler and John Nash did for example).
> > > >
> > > > The solution to this is to find study methods that work FOR YOU (which might not be the same as what works for someone else), and which books work best FOR YOU. I will give what I hope is a helpful list for both at the end of this post, including some genuinely useful YouTube channels. Once you find material and approaches that work for you, you can be surprised at how much more smoothly things go; but it is never truly easy, because maths is abstract and often subtle, and abstraction is always difficult.
> > > >
> > > > And it is perfectly normal for "philosophical" questions about a subject to sometimes occur to students, which they might find difficult to verbalize, and might not feel their book is directly addressing.
> > > >
> > > > Under no circumstances is the right response to this, to seek refuge in silly ancient Greek metaphysics, amateur psychology, and fantasizing that 150 years worth of mathematics experts missed the real value of an ancient Greek geometry book that had been regularly read and understood for centuries; nor that they are clueless about set theory that has been thought about for 150 years. Philosophers and mathematicians of many cultures have been thinking about abstract ideas of number and space for millennia; Mr Gabriel's ideas are neither deep, nor special, nor generally effective.
> > > >
> > > > Qualified mathematicians and teachers, including myself, have read Mr Gabriel's New Calculus and are not impressed by it. It can work in a very small class of situations, just like other pre-limit methods which existed in the 16th century, but beyond that not really. It also struggles with basic and important ideas like points of inflection, which are important in pure mathematics, game theory, the calculus of variations and other areas.
> > > >
> > > > Some examples of this kind of historical development can be found here, in this lovely 25 minute 1980's BBC/Open University video starring mathematics historian Jeremy Gray:
> > > >
> > > > https://www.youtube.com/watch?v=ObPg3ki9GOI
> > > >
> > > > Needless to say, Mr Gabriel's approach cannot be used to develop a useful calculus of variations nor indeed many of the standard methods of modern mathematical physics.
> > > >
> > > > Mr Gabriel is also mistaken in believing that he has a special insight into the concept of number. Elementary conceptions of number were common in higher mathematics pre-20th century. Leonhard Euler gave a much smoother and fruitful offering of one in his lovely book:
> > > >
> > > > "Elements of Algebra"
> > > >
> > > > which has been read since the 18th century, and can still be purchased in various abridged forms today. Euler was a genius and clear explainer, but no one (especially not Euler himself) claims he was a genius because of his smooth presentation of a number concept.
> > > >
> > > > POSITIVE RECOMMENDATIONS FOR STUDENTS
> > > >
> > > > 1] Concerning philosophy
> > > >
> > > > Most philosophical difficulties with early set theory have been satisfactorily addressed, but the detailed explanations can be very hard and dwelling on them during undergraduate studies is often not advisable because it can take far too long, and is better approached when you have more experience.
> > > >
> > > > There is nothing wrong with having philosophical opinions that differ from the mainstream per se, provided they are not dogmatic and provided they possess clarity. They should be clearly thought out, should not inhibit your ability to do core mathematics (algebra, calculus etc.) and shouldn't descend into silly conspiracy theories about academia.
> > > >
> > > > If, for example, students feel drawn to a more algorithmic and less Cantorian approach to mathematics than is normally presented, then I would suggest they look up Professor Harold M. Edwards, who has championed an algorithmic approach to mathematics for many years. His webpage is here:
> > > >
> > > > https://math.nyu.edu/faculty/edwardsd/
> > > >
> > > > In particular his book on elementary number theory:
> > > >
> > > > Higher Arithmetic: an algorithmic introduction to number theory
> > > >
> > > > might be suitable for such students. I do not share Professor Edwards' philosophical opinions, but at least he is intellectually sober and does genuinely good mathematics within the framework he advocates. So if you dislike or struggle with Cantorianism, then try Edwards instead, don't waste your time with vague ancient Greek metaphysics and ultra-finitist nonsense..
> > > >
> > > >
> > > > 2] On psychology and learning
> > > >
> > > > Psychologists have been studying the learning of mathematics for decades, and some genuinely useful advice for studying more effectively has come out of it. Look up:
> > > >
> > > > - spaced practice and retrieval practice
> > > >
> > > > - Frayer models/diagrams
> > > >
> > > > - the brilliant lecture: https://www.youtube.com/watch?v=IlU-zDU6aQ0&t
> > > >
> > > > - the benefits of cardio-vascular exercise for improving study focus (the neurologist John Ratey has presented good stuff on this kind of thing)
> > > >
> > > > - The psychologist Anders Ericsson, who researched the development of expertise.
> > > >
> > > > Don't fall into the trap of thinking that if you stare at a problem for a few minutes and can't do it, that this means you aren't able to be very good at the subject. Many mathematicians need to ponder things at length and return to the same thing in an iterative fashion over a period of time. Even genius's like Nobel prize winning mathematical physicist Roger Penrose do this, and if you want to see what slow, fallible pondering can produce, look up his book:
> > > >
> > > > The Road to Reality
> > > >
> > > > which provides beautiful illustrations of what many of the mathematical ideas you study can be used to do in physics. You won't find a better guide than Penrose on that sort of thing. A brilliant teacher and a true inspiration.
> > > >
> > > > The book: How to Solve it, by G. Polya, is a classic short guide for undergraduates in developing problem solving skills in mathematics. Far more worthy of your time than Mr Gabriel's output.
> > > >
> > > >
> > > > 3] Books on analysis
> > > >
> > > > The best book on analysis FOR YOU, is the one that is easiest to learn from FOR YOU. There is no universally suitable book and no book is perfect. An excellent book for students who lack confidence is:
> > > >
> > > > Fundamentals of Mathematical Analysis 2nd ed, by Rod Haggarty.
> > > >
> > > > It goes at a gentle pace, has diagrams, historical asides, and solutions to exercises. Good for self-study. It's only significant drawback is that it doesn't construct the real numbers. But for that, you can look up explanations of Dedekind cuts on the internet.
> > > >
> > > > I found my first and second year courses on real analysis quite difficult at first, but I bought Haggarty's book, threw myself into it and I aced my courses and went on to study the Calculus of Variations at MSc level later on. I didn't disrupt my education by indulging the silly idea that real analysis was built on obvious delusions and lies. That way represents the slow decent into bitterness and madness.
> > > >
> > > > A book which is somewhat more difficult (but not impossibly hard), but which has a beautiful style of exposition and does look at Dedekind cuts, is the classic text:
> > > >
> > > > A Course in Pure Mathematics (centenary edition), by G.H. Hardy
> > > >
> > > > Having both texts so that you can switch between them is a perfectly good strategy. Don't be afraid to have different books on the same subject. Professional mathematicians often do.
> > > >
> > > >
> > > > 4] Genuinely helpful YouTube channels
> > > >
> > > > One of the least toxic places on YouTube, hosted by a university lecturer, lots of short advice videos about studying mathematics, book reviews and much else:
> > > >
> > > > The Math Sorcerer: https://www.youtube.com/user/themathsorcerer
> > > >
> > > > For good lecture series on mathematics and other topics, such as linear algebra, visit MITOpenCourseWare
> > > >
> > > > https://www.youtube.com/channel/UCEBb1b_L6zDS3xTUrIALZOw
> > > >
> > > > For some very beautiful visual presentations of mathematical ideas, you would find it hard to do better than 3Blue1Brown:
> > > >
> > > > https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
> > > >
> > > > Best of luck with your mathematics studies, and don't allow yourselves to be done a disservice by silly conspiracy theories. Seek assistance from your college or university if you are having difficulty with a particular subject or study in general; it is perfectly normal, and can be dealt with by seeking out good advice from your tutors, library or study support departments. Not by wasting your time on delirious rantings on the internet.
> > > >
> > > > Kindest Regards
> > >
> > > This lying anonymous bastard continues to spew out his vomit.
> > >
> > > Read for yourself what I have to say and not what some dumb UK cunt spews out here on sci.math:
> > >
> > > https://independent.academia.edu/JohnGabriel30
> > Begin with my historic geometric theorem of January 2020 which exposes the lie that mainstream calculus was ever rigorous:
> >
> > https://www.academia.edu/62358358/My_historic_geometric_theorem_of_January_2020
> >
> > https://www.academia.edu/79881709/Six_simple_reasons_why_the_mainstream_derivative_definition_of_calculus_is_flawed
> >
> > and much more at the above link: https://independent.academia.edu/JohnGabriel30
> Want to learn calculus in just two pages? See below.
>
> https://www.academia.edu/103723139/My_historic_geometric_theorem_for_Dummies

måndag 26 juni 2023 kl. 17:39:27 UTC+2 skrev Eram semper recta:
> On Thursday, 11 May 2023 at 14:44:20 UTC-4, Eram semper recta wrote:
> > On Thursday, 11 May 2023 at 14:41:56 UTC-4, Eram semper recta wrote:
> > > On Tuesday, 13 April 2021 at 06:23:36 UTC-4, Quantum Bubbles wrote:
> > > > Mr Gabriel's "New Calculus" will not give you an advantage in understanding mathematics and will stall your mathematical development if you take it too seriously. As someone with multiple mathematics degrees and a teaching qualification in mathematics, I am genuinely disturbed by the idea that students, perhaps suffering from anxiety, lack of self-confidence and other very common difficulties, might have their precious time and effort wasted by this.
> > > >
> > > > Calculus and analysis can be tricky subjects to learn, and even trickier to master. There is a lot of material to digest; many definitions, many techniques, the calculus books are sometimes 1000 pages long, and the analysis books seem to be packed with a blizzard of inequalities. You might have spent days learning an idea and then weeks later it feels as if you have forgotten it all.
> > > >
> > > > It can look very daunting. This is a perfectly normal feeling and neither signifies that you cannot do it, nor that the mathematics community does not know what it is doing. There are many books on the same subjects, and different students will be better served by different choices of books, or different ways of presenting the same ideas. Teachers cannot always cater for every preference at once; but that does not mean there isn't a good book FOR YOU out there.
> > > >
> > > > It is perfectly normal for it to take years to master these topics, and it is perfectly normal to struggle at various stages. It is normal to struggle at the beginning when a mathematical concept is new. It is normal to struggle when the topics have become more abstract. It is normal to struggle when connections between different ideas need to be explored, because the cognitive load is demanding. It is normal for memories to fade after the pressure of exams has ended, unless you practice to help solidify those memories afterwards.
> > > >
> > > > This is true for most people; it doesn't really matter which university you go to, or how smart you are (even genius's sometimes need feedback from other experts: Leonhard Euler and John Nash did for example).
> > > >
> > > > The solution to this is to find study methods that work FOR YOU (which might not be the same as what works for someone else), and which books work best FOR YOU. I will give what I hope is a helpful list for both at the end of this post, including some genuinely useful YouTube channels. Once you find material and approaches that work for you, you can be surprised at how much more smoothly things go; but it is never truly easy, because maths is abstract and often subtle, and abstraction is always difficult.
> > > >
> > > > And it is perfectly normal for "philosophical" questions about a subject to sometimes occur to students, which they might find difficult to verbalize, and might not feel their book is directly addressing.
> > > >
> > > > Under no circumstances is the right response to this, to seek refuge in silly ancient Greek metaphysics, amateur psychology, and fantasizing that 150 years worth of mathematics experts missed the real value of an ancient Greek geometry book that had been regularly read and understood for centuries; nor that they are clueless about set theory that has been thought about for 150 years. Philosophers and mathematicians of many cultures have been thinking about abstract ideas of number and space for millennia; Mr Gabriel's ideas are neither deep, nor special, nor generally effective.
> > > >
> > > > Qualified mathematicians and teachers, including myself, have read Mr Gabriel's New Calculus and are not impressed by it. It can work in a very small class of situations, just like other pre-limit methods which existed in the 16th century, but beyond that not really. It also struggles with basic and important ideas like points of inflection, which are important in pure mathematics, game theory, the calculus of variations and other areas.
> > > >
> > > > Some examples of this kind of historical development can be found here, in this lovely 25 minute 1980's BBC/Open University video starring mathematics historian Jeremy Gray:
> > > >
> > > > https://www.youtube.com/watch?v=ObPg3ki9GOI
> > > >
> > > > Needless to say, Mr Gabriel's approach cannot be used to develop a useful calculus of variations nor indeed many of the standard methods of modern mathematical physics.
> > > >
> > > > Mr Gabriel is also mistaken in believing that he has a special insight into the concept of number. Elementary conceptions of number were common in higher mathematics pre-20th century. Leonhard Euler gave a much smoother and fruitful offering of one in his lovely book:
> > > >
> > > > "Elements of Algebra"
> > > >
> > > > which has been read since the 18th century, and can still be purchased in various abridged forms today. Euler was a genius and clear explainer, but no one (especially not Euler himself) claims he was a genius because of his smooth presentation of a number concept.
> > > >
> > > > POSITIVE RECOMMENDATIONS FOR STUDENTS
> > > >
> > > > 1] Concerning philosophy
> > > >
> > > > Most philosophical difficulties with early set theory have been satisfactorily addressed, but the detailed explanations can be very hard and dwelling on them during undergraduate studies is often not advisable because it can take far too long, and is better approached when you have more experience.
> > > >
> > > > There is nothing wrong with having philosophical opinions that differ from the mainstream per se, provided they are not dogmatic and provided they possess clarity. They should be clearly thought out, should not inhibit your ability to do core mathematics (algebra, calculus etc.) and shouldn't descend into silly conspiracy theories about academia.
> > > >
> > > > If, for example, students feel drawn to a more algorithmic and less Cantorian approach to mathematics than is normally presented, then I would suggest they look up Professor Harold M. Edwards, who has championed an algorithmic approach to mathematics for many years. His webpage is here:
> > > >
> > > > https://math.nyu.edu/faculty/edwardsd/
> > > >
> > > > In particular his book on elementary number theory:
> > > >
> > > > Higher Arithmetic: an algorithmic introduction to number theory
> > > >
> > > > might be suitable for such students. I do not share Professor Edwards' philosophical opinions, but at least he is intellectually sober and does genuinely good mathematics within the framework he advocates. So if you dislike or struggle with Cantorianism, then try Edwards instead, don't waste your time with vague ancient Greek metaphysics and ultra-finitist nonsense..
> > > >
> > > >
> > > > 2] On psychology and learning
> > > >
> > > > Psychologists have been studying the learning of mathematics for decades, and some genuinely useful advice for studying more effectively has come out of it. Look up:
> > > >
> > > > - spaced practice and retrieval practice
> > > >
> > > > - Frayer models/diagrams
> > > >
> > > > - the brilliant lecture: https://www.youtube.com/watch?v=IlU-zDU6aQ0&t
> > > >
> > > > - the benefits of cardio-vascular exercise for improving study focus (the neurologist John Ratey has presented good stuff on this kind of thing)
> > > >
> > > > - The psychologist Anders Ericsson, who researched the development of expertise.
> > > >
> > > > Don't fall into the trap of thinking that if you stare at a problem for a few minutes and can't do it, that this means you aren't able to be very good at the subject. Many mathematicians need to ponder things at length and return to the same thing in an iterative fashion over a period of time. Even genius's like Nobel prize winning mathematical physicist Roger Penrose do this, and if you want to see what slow, fallible pondering can produce, look up his book:
> > > >
> > > > The Road to Reality
> > > >
> > > > which provides beautiful illustrations of what many of the mathematical ideas you study can be used to do in physics. You won't find a better guide than Penrose on that sort of thing. A brilliant teacher and a true inspiration.
> > > >
> > > > The book: How to Solve it, by G. Polya, is a classic short guide for undergraduates in developing problem solving skills in mathematics. Far more worthy of your time than Mr Gabriel's output.
> > > >
> > > >
> > > > 3] Books on analysis
> > > >
> > > > The best book on analysis FOR YOU, is the one that is easiest to learn from FOR YOU. There is no universally suitable book and no book is perfect. An excellent book for students who lack confidence is:
> > > >
> > > > Fundamentals of Mathematical Analysis 2nd ed, by Rod Haggarty.
> > > >
> > > > It goes at a gentle pace, has diagrams, historical asides, and solutions to exercises. Good for self-study. It's only significant drawback is that it doesn't construct the real numbers. But for that, you can look up explanations of Dedekind cuts on the internet.
> > > >
> > > > I found my first and second year courses on real analysis quite difficult at first, but I bought Haggarty's book, threw myself into it and I aced my courses and went on to study the Calculus of Variations at MSc level later on. I didn't disrupt my education by indulging the silly idea that real analysis was built on obvious delusions and lies. That way represents the slow decent into bitterness and madness.
> > > >
> > > > A book which is somewhat more difficult (but not impossibly hard), but which has a beautiful style of exposition and does look at Dedekind cuts, is the classic text:
> > > >
> > > > A Course in Pure Mathematics (centenary edition), by G.H. Hardy
> > > >
> > > > Having both texts so that you can switch between them is a perfectly good strategy. Don't be afraid to have different books on the same subject. Professional mathematicians often do.
> > > >
> > > >
> > > > 4] Genuinely helpful YouTube channels
> > > >
> > > > One of the least toxic places on YouTube, hosted by a university lecturer, lots of short advice videos about studying mathematics, book reviews and much else:
> > > >
> > > > The Math Sorcerer: https://www.youtube.com/user/themathsorcerer
> > > >
> > > > For good lecture series on mathematics and other topics, such as linear algebra, visit MITOpenCourseWare
> > > >
> > > > https://www.youtube.com/channel/UCEBb1b_L6zDS3xTUrIALZOw
> > > >
> > > > For some very beautiful visual presentations of mathematical ideas, you would find it hard to do better than 3Blue1Brown:
> > > >
> > > > https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
> > > >
> > > > Best of luck with your mathematics studies, and don't allow yourselves to be done a disservice by silly conspiracy theories. Seek assistance from your college or university if you are having difficulty with a particular subject or study in general; it is perfectly normal, and can be dealt with by seeking out good advice from your tutors, library or study support departments. Not by wasting your time on delirious rantings on the internet.
> > > >
> > > > Kindest Regards
> > >
> > > This lying anonymous bastard continues to spew out his vomit.
> > >
> > > Read for yourself what I have to say and not what some dumb UK cunt spews out here on sci.math:
> > >
> > > https://independent.academia.edu/JohnGabriel30
> > Begin with my historic geometric theorem of January 2020 which exposes the lie that mainstream calculus was ever rigorous:
> >
> > https://www.academia.edu/62358358/My_historic_geometric_theorem_of_January_2020
> >
> > https://www.academia.edu/79881709/Six_simple_reasons_why_the_mainstream_derivative_definition_of_calculus_is_flawed
> >
> > and much more at the above link: https://independent.academia.edu/JohnGabriel30
> Want to learn calculus in just two pages? See below.
>
> https://www.academia.edu/103723139/My_historic_geometric_theorem_for_Dummies
You still struggle with calculus?

On Thursday, 7 September 2023 at 12:28:55 UTC-4, markus...
> > > > >
> > > > > Calculus and analysis can be tricky subjects to learn, and even trickier to master. There is a lot of material to digest; many definitions, many techniques, the calculus books are sometimes 1000 pages long, and the analysis books seem to be packed with a blizzard of inequalities. You might have spent days learning an idea and then weeks later it feels as if you have forgotten it all.
> > > > >
> > > > > It can look very daunting. This is a perfectly normal feeling and neither signifies that you cannot do it, nor that the mathematics community does not know what it is doing. There are many books on the same subjects, and different students will be better served by different choices of books, or different ways of presenting the same ideas. Teachers cannot always cater for every preference at once; but that does not mean there isn't a good book FOR YOU out there.
> > > > >
> > > > > It is perfectly normal for it to take years to master these topics, and it is perfectly normal to struggle at various stages. It is normal to struggle at the beginning when a mathematical concept is new. It is normal to struggle when the topics have become more abstract. It is normal to struggle when connections between different ideas need to be explored, because the cognitive load is demanding. It is normal for memories to fade after the pressure of exams has ended, unless you practice to help solidify those memories afterwards.
> > > > >
> > > > > This is true for most people; it doesn't really matter which university you go to, or how smart you are (even genius's sometimes need feedback from other experts: Leonhard Euler and John Nash did for example).
> > > > >
> > > > > The solution to this is to find study methods that work FOR YOU (which might not be the same as what works for someone else), and which books work best FOR YOU. I will give what I hope is a helpful list for both at the end of this post, including some genuinely useful YouTube channels. Once you find material and approaches that work for you, you can be surprised at how much more smoothly things go; but it is never truly easy, because maths is abstract and often subtle, and abstraction is always difficult.
> > > > >
> > > > > And it is perfectly normal for "philosophical" questions about a subject to sometimes occur to students, which they might find difficult to verbalize, and might not feel their book is directly addressing.
> > > > >
> > > > > Under no circumstances is the right response to this, to seek refuge in silly ancient Greek metaphysics, amateur psychology, and fantasizing that 150 years worth of mathematics experts missed the real value of an ancient Greek geometry book that had been regularly read and understood for centuries; nor that they are clueless about set theory that has been thought about for 150 years. Philosophers and mathematicians of many cultures have been thinking about abstract ideas of number and space for millennia; Mr Gabriel's ideas are neither deep, nor special, nor generally effective.
> > > > >
> > > > > Qualified mathematicians and teachers, including myself, have read Mr Gabriel's New Calculus and are not impressed by it. It can work in a very small class of situations, just like other pre-limit methods which existed in the 16th century, but beyond that not really. It also struggles with basic and important ideas like points of inflection, which are important in pure mathematics, game theory, the calculus of variations and other areas..
> > > > >
> > > > > Some examples of this kind of historical development can be found here, in this lovely 25 minute 1980's BBC/Open University video starring mathematics historian Jeremy Gray:
> > > > >
> > > > > https://www.youtube.com/watch?v=ObPg3ki9GOI
> > > > >
> > > > > Needless to say, Mr Gabriel's approach cannot be used to develop a useful calculus of variations nor indeed many of the standard methods of modern mathematical physics.
> > > > >
> > > > > Mr Gabriel is also mistaken in believing that he has a special insight into the concept of number. Elementary conceptions of number were common in higher mathematics pre-20th century. Leonhard Euler gave a much smoother and fruitful offering of one in his lovely book:
> > > > >
> > > > > "Elements of Algebra"
> > > > >
> > > > > which has been read since the 18th century, and can still be purchased in various abridged forms today. Euler was a genius and clear explainer, but no one (especially not Euler himself) claims he was a genius because of his smooth presentation of a number concept.
> > > > >
> > > > > POSITIVE RECOMMENDATIONS FOR STUDENTS
> > > > >
> > > > > 1] Concerning philosophy
> > > > >
> > > > > Most philosophical difficulties with early set theory have been satisfactorily addressed, but the detailed explanations can be very hard and dwelling on them during undergraduate studies is often not advisable because it can take far too long, and is better approached when you have more experience.
> > > > >
> > > > > There is nothing wrong with having philosophical opinions that differ from the mainstream per se, provided they are not dogmatic and provided they possess clarity. They should be clearly thought out, should not inhibit your ability to do core mathematics (algebra, calculus etc.) and shouldn't descend into silly conspiracy theories about academia.
> > > > >
> > > > > If, for example, students feel drawn to a more algorithmic and less Cantorian approach to mathematics than is normally presented, then I would suggest they look up Professor Harold M. Edwards, who has championed an algorithmic approach to mathematics for many years. His webpage is here:
> > > > >
> > > > > https://math.nyu.edu/faculty/edwardsd/
> > > > >
> > > > > In particular his book on elementary number theory:
> > > > >
> > > > > Higher Arithmetic: an algorithmic introduction to number theory
> > > > >
> > > > > might be suitable for such students. I do not share Professor Edwards' philosophical opinions, but at least he is intellectually sober and does genuinely good mathematics within the framework he advocates. So if you dislike or struggle with Cantorianism, then try Edwards instead, don't waste your time with vague ancient Greek metaphysics and ultra-finitist nonsense.
> > > > >
> > > > >
> > > > > 2] On psychology and learning
> > > > >
> > > > > Psychologists have been studying the learning of mathematics for decades, and some genuinely useful advice for studying more effectively has come out of it. Look up:
> > > > >
> > > > > - spaced practice and retrieval practice
> > > > >
> > > > > - Frayer models/diagrams
> > > > >
> > > > > - the brilliant lecture: https://www.youtube.com/watch?v=IlU-zDU6aQ0&t
> > > > >
> > > > > - the benefits of cardio-vascular exercise for improving study focus (the neurologist John Ratey has presented good stuff on this kind of thing)
> > > > >
> > > > > - The psychologist Anders Ericsson, who researched the development of expertise.
> > > > >
> > > > > Don't fall into the trap of thinking that if you stare at a problem for a few minutes and can't do it, that this means you aren't able to be very good at the subject. Many mathematicians need to ponder things at length and return to the same thing in an iterative fashion over a period of time. Even genius's like Nobel prize winning mathematical physicist Roger Penrose do this, and if you want to see what slow, fallible pondering can produce, look up his book:
> > > > >
> > > > > The Road to Reality
> > > > >
> > > > > which provides beautiful illustrations of what many of the mathematical ideas you study can be used to do in physics. You won't find a better guide than Penrose on that sort of thing. A brilliant teacher and a true inspiration.
> > > > >
> > > > > The book: How to Solve it, by G. Polya, is a classic short guide for undergraduates in developing problem solving skills in mathematics. Far more worthy of your time than Mr Gabriel's output.
> > > > >
> > > > >
> > > > > 3] Books on analysis
> > > > >
> > > > > The best book on analysis FOR YOU, is the one that is easiest to learn from FOR YOU. There is no universally suitable book and no book is perfect. An excellent book for students who lack confidence is:
> > > > >
> > > > > Fundamentals of Mathematical Analysis 2nd ed, by Rod Haggarty.
> > > > >
> > > > > It goes at a gentle pace, has diagrams, historical asides, and solutions to exercises. Good for self-study. It's only significant drawback is that it doesn't construct the real numbers. But for that, you can look up explanations of Dedekind cuts on the internet.
> > > > >
> > > > > I found my first and second year courses on real analysis quite difficult at first, but I bought Haggarty's book, threw myself into it and I aced my courses and went on to study the Calculus of Variations at MSc level later on. I didn't disrupt my education by indulging the silly idea that real analysis was built on obvious delusions and lies. That way represents the slow decent into bitterness and madness.
> > > > >
> > > > > A book which is somewhat more difficult (but not impossibly hard), but which has a beautiful style of exposition and does look at Dedekind cuts, is the classic text:
> > > > >
> > > > > A Course in Pure Mathematics (centenary edition), by G.H. Hardy
> > > > >
> > > > > Having both texts so that you can switch between them is a perfectly good strategy. Don't be afraid to have different books on the same subject. Professional mathematicians often do.
> > > > >
> > > > >
> > > > > 4] Genuinely helpful YouTube channels
> > > > >
> > > > > One of the least toxic places on YouTube, hosted by a university lecturer, lots of short advice videos about studying mathematics, book reviews and much else:
> > > > >
> > > > > The Math Sorcerer: https://www.youtube.com/user/themathsorcerer
> > > > >
> > > > > For good lecture series on mathematics and other topics, such as linear algebra, visit MITOpenCourseWare
> > > > >
> > > > > https://www.youtube.com/channel/UCEBb1b_L6zDS3xTUrIALZOw
> > > > >
> > > > > For some very beautiful visual presentations of mathematical ideas, you would find it hard to do better than 3Blue1Brown:
> > > > >
> > > > > https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
> > > > >
> > > > > Best of luck with your mathematics studies, and don't allow yourselves to be done a disservice by silly conspiracy theories. Seek assistance from your college or university if you are having difficulty with a particular subject or study in general; it is perfectly normal, and can be dealt with by seeking out good advice from your tutors, library or study support departments. Not by wasting your time on delirious rantings on the internet..
> > > > >
> > > > > Kindest Regards
> > > >
> > > > This lying anonymous bastard continues to spew out his vomit.
> > > >
> > > > Read for yourself what I have to say and not what some dumb UK cunt spews out here on sci.math:
> > > >
> > > > https://independent.academia.edu/JohnGabriel30
> > > Begin with my historic geometric theorem of January 2020 which exposes the lie that mainstream calculus was ever rigorous:
> > >
> > > https://www.academia.edu/62358358/My_historic_geometric_theorem_of_January_2020
> > >
> > > https://www.academia.edu/79881709/Six_simple_reasons_why_the_mainstream_derivative_definition_of_calculus_is_flawed
> > >
> > > and much more at the above link: https://independent.academia.edu/JohnGabriel30
> > Want to learn calculus in just two pages? See below.
> >
> > https://www.academia.edu/103723139/My_historic_geometric_theorem_for_Dummies
> You still struggle with calculus?