7 ms·

What is it like to understand advanced mathematics?

122 points by vinchuco 9y ago

moocowtruck 9y ago
to understand advanced mathematics, is to start reading books on philosophy
- KGIII 9y ago
  Pretty much. Mathematics is a language. Most people only learn arithmetic, and then only by rote. Sometimes, it is disheartening. I've had people who refused to accept that Ph.D. is Doctor of Philosophy. I like to point out that nobody has successfully brought me a bucket of negative apples. Err... I'm a mathematician. I am an applied mathematician. I am not smart enough to be a philosopher of mathematics, though I have my Ph.D. Anyhow, math is a language that is capable of telling stories, truths, and even falsehoods.
  - dgut 9y ago
    That PhD means "Doctor of Philosophy" is just an artifact of the fact that all doctoral degrees were once subcategories of "Philosophy". Philosophy today is pretty useless.
    - pmoriarty 9y ago
      "Philosophy today is pretty useless." Studying philosophy teaches you how to think. If you consider that useless...
      - waqf 9y ago
        How do you know it doesn't just draw out a capacity to think that you already have, and that you could equally well have drawn out by studying or working at something else?
        pmoriarty 9y ago
        Maybe you could, but that still doesn't make philosophy useless.
      - webnrrd2k 9y ago
        If philosophy (and only philosophy) teaches people how to think, then who taught the first philosopher?
        pmoriarty 9y ago
        No one claimed only philosophy taught people how to think.
      - dgut 9y ago
        You claim that. Yet none of the most mentally demanding tasks were done by philosophers. Today philosophy is mostly meaningless intellectual posturing. In the past, it was the act of thinking without applying the scientific method. As Feynman said of Spinoza "meaningless chewing around".
    - muninn_ 9y ago
      I think it's sad that your view that philosophy is worthless has become more of a mainstream feeling than it ought to be. Not everything needs to be measured in terms of usefulness. Interestingly enough, I think you find many developers have a natural affinity for philosophy. All this talk about AI, for example, is just philosophy.
    - pcmaffey 9y ago
      Most philosophers throughout history would laugh at their body of knowledge being measured in terms of utility. And then would become saddened that economics has become the dominant global value system.
      - notduncansmith 9y ago
        Economics is the word for the dominant global value system. I think a sad philosopher has more work to do.
    - pjmorris 9y ago
      I'd argue that philosophy may become more important than ever as we contemplate the ethics of handing decisions like who's going to die in a trolley-problem type self-driving car accident over to machines.
      - maxerickson 9y ago
        It'll be easier to (metaphorically) untie people from the tracks. Program slow speed limits in dangerous areas, install sensor augmentation in areas where vehicle located sensors don't work well, rebuild roads to work better, etc.
    - Ixiaus 9y ago
      A bit parochial to say that Philosophy--as it is taught or studied today--is useless, in my opinion. You're entitled to that thought but you might be surprised by the utility Philosophy has to offer. I think Philosophy studied with other disciplines is a particularly potent combination. Of my friends who have done this (they are very few) they are perhaps the most inspiring and trenchant thinkers I know.
  - Jtsummers 9y ago
    No one would bring you a bucket of negative apples. Positive and negative ought to be thought of as directions, while the absolute value is the magnitude. A bucket of apples given to you has a positive direction (increasing your supply). A bucket of apples being taken from you has a negative direction (decreasing your supply).
    - KGIII 9y ago
      Of course you can't bring me a bucket of negative apples. That's the point. It very much is a philosophy and doesn't always relate to reality. Reality is, you can never really have a bucket of negative apples.
      - vinchuco 9y ago
        But it's your fault no one can bring you this bucket: You want 'negative apple' entities but ask for (what is interpreted as) a negative number of apple entities. A key distinction here is that one has a defined implementation. Making that distinction is philosophy. Talking within the framework of implementation is mathematics.
        KGIII 9y ago
        Right. It's very much a philosophy. And it's not really my fault that nobody can bring me negative apples. It's not like I made the rules! Sheesh! ;-)
        vinchuco 9y ago
        You did! You posed the question! I just don't see the value in conflating both terms (philos and math) when there's a distinction pointed out above.
        KGIII 9y ago
        No, I didn't make the rules. Reality is what reality is. They don't let me decide the rules for reality. That's probably for the best. You can't bring me a bucket of negative apples. You can bring representations of future apples, or something like that. However, negative apples do not actually exist. In all these years, nobody has brought me a basket of negative apples.
  - smohare 9y ago
    As another fellow mathematician (albeit pure), I take some issue with your argument. Rather, I'd say the collection of apples is a clear manifestation of a monoid that is not a group. Why should all physical systems form a group? It seems almost intellectually dishonest to suggest that a particular non-example is evidence against the universality of a mathematical object. Are symmetry groups not "physical" enough? I can rotate a piece of paper in the plane and then undo this operation just fine. Algebraic and topological structures seem to manifest themselves readily within nature. I should think infinities provide greater concern.
    - deleted 9y ago
      [deleted]
    - KGIII 9y ago
      The post may not have been clear enough, but the point is that it's impossible to really have negative apples. You can owe apples, but you can't physically possess them. Namely, pointing out how it remains a philosophical pursuit. As for infinities... I'm not sure if they drove Cantor insane or if he was already insane. Some infinities are bigger than others, and I have strong opinions about infinity. 0.999... = 1 Horrible! ;-)
      - BoiledCabbage 9y ago
        I disagree, I can physically possess 3 apples no more nor less than I can physically possess -3 apples. To say "I have three apples" is merely a way of modeling a configuration of the world. The apples may be in my pocket, or in my hand, or in a bag I'm holding, or even on my desk in front of me. They may be in the barn out back, but in all cases we can say I have 3 apples. Saying "i have 3 apples" is a simplification of a ton of configurations of the state of the world where I have ownership of apples. Similarly I have -3 apples is a summary of all the configurations of the world where I owe three apples. Neither is any more real than the other. And yes it is philosophical.
    - smnplk 9y ago
      >> Algebraic and topological structures seem to manifest themselves readily within nature. I should think infinities provide greater concern. So basically what you are saying is that, we can't be so certain that we humans invented math. Math is there, structures were there to be discovered ? Why is there such a thing as monster group ? We named those things, but if we didn't exists, it would still be there in the abstract world ? This is what is bugging me lately, I can't decide if math was invented or not. Some parts may be invented, but some parts may be discovered.
  - SilasX 9y ago
    Not sure what that remark was in reference to, but ... yes, they can bring you negative apples. Your enterprise can have an obligation to deliver apples to someone, which has all the same effects as negative apples and should be booked as such. If you acquired an enterprise with such an obligation, they gave you negative apples. It's important to account for them that way in e.g. seeing the implications of the Put-Call Parity Theroem.
    - KGIII 9y ago
      See, there's no such thing as negative apples. Sure, they exist in math - but they don't exist in reality. You can possess a chit that says you owe apples but, in reality, you only have said chit and do not actually possess negative apples. The post may not have been clear enough, but the point was to show that it's still very much a philosophy.
  - smegel 9y ago
    > nobody has successfully brought me a bucket of negative apples. But when they did I am going to take the square root of the number of them and break the universe.
    - vinchuco 9y ago
      I should have thought of this sooner. Since they're ALREADY talking about purely imaginary apples, take n of these. Arrange them in the bucket so as to make a square of side length n (of said imaginary apples). The result is n^2 negative apples and a really bad math joke.
  - goodbye- 9y ago
    It must be extremely difficult for you to drive around in your car that is unable to go in reverse. Math is fascinating. It is also just a tool.
- vijayr 9y ago
  For someone who has forgotten math (other than high school level) and never did philosophy, what books/courses are good to start with (especially philosophy)?
  - subwayclub 9y ago
    The community college course I took past spring used Thomas Nagel (What Does It All Mean?) supplemented with some handouts. I would recommend a class because part of the doing of philosophy is the discussion. Plus the classroom is one of the few places where you can delve into the nuances of an issue without being considered pedantic or boorish.
swayvil 9y ago
It's like being really good at a video game that only you and maybe 2 other guys play. You talk about the dungeons, monsters, level-bosses, etc but nobody can relate.
- deleted 9y ago
  [deleted]
paultopia 9y ago
Interestingly, this reads like the experience of expert knowledge in almost anything. Like, I have expert knowledge in law, and I'm working toward at least more advanced (though far far from expert) knowledge in coding-things, and it very nicely matches how I interact with legal problems and can dimly see glimmers off at the end of a long tunnel in coding problems. But at the same time, it's inspiring as all hell, because mathematics from the outside doesn't seem like the kind of thing that could ever produce the same satisfying and effective experiences of cognition.
nopinsight 9y ago
I somehow feel like this is written by Terence Tao, based on the quality, style, and length that are similar to what's on his blog. Anyone else shares the feeling or has a different guess? Edit: In a later part, the answer says "Terence Tao is very eloquent about this here: ..." so likely not (based on factors like typical personality of a mathematician, but I could be wrong since I don't know him personally).
- deleted 9y ago
  [deleted]
- vinchuco 9y ago
  https://en.wikipedia.org/wiki/Stylometry https://en.wikipedia.org/wiki/Stylometry
- HiroshiSan 9y ago
  that later part could be a nice red herring for those who think it is Terry.
jzl 9y ago
There are many wonderful passages in the book "Genius" by James Gleick that describe Richard Feynman's deep intuitive understanding of calculus. He also talks about it a bit in his own works such as Surely You're Joking Mr. Feynman. Highly recommend if you want to see lots of interesting answers to this question. His key approach was that he always wanted to understand things based on "first principles". So he would usually take on a new mathematical frameworks by breaking them down and re-deriving them on his own from scratch. Reminds me of how Ben Franklin taught himself to write by reading famous passages and then attempting to write them from memory, comparing his version to the original version afterward and seeing what he did differently and why.

122 points by vinchuco 9y ago

moocowtruck 9y ago

to understand advanced mathematics, is to start reading books on philosophy

KGIII 9y ago
Pretty much. Mathematics is a language. Most people only learn arithmetic, and then only by rote. Sometimes, it is disheartening. I've had people who refused to accept that Ph.D. is Doctor of Philosophy. I like to point out that nobody has successfully brought me a bucket of negative apples. Err... I'm a mathematician. I am an applied mathematician. I am not smart enough to be a philosopher of mathematics, though I have my Ph.D. Anyhow, math is a language that is capable of telling stories, truths, and even falsehoods.
- dgut 9y ago
  That PhD means "Doctor of Philosophy" is just an artifact of the fact that all doctoral degrees were once subcategories of "Philosophy". Philosophy today is pretty useless.
  - pmoriarty 9y ago
    "Philosophy today is pretty useless." Studying philosophy teaches you how to think. If you consider that useless...
    - waqf 9y ago
      How do you know it doesn't just draw out a capacity to think that you already have, and that you could equally well have drawn out by studying or working at something else?
      - pmoriarty 9y ago
        Maybe you could, but that still doesn't make philosophy useless.
    - webnrrd2k 9y ago
      If philosophy (and only philosophy) teaches people how to think, then who taught the first philosopher?
      - pmoriarty 9y ago
        No one claimed only philosophy taught people how to think.
    - dgut 9y ago
      You claim that. Yet none of the most mentally demanding tasks were done by philosophers. Today philosophy is mostly meaningless intellectual posturing. In the past, it was the act of thinking without applying the scientific method. As Feynman said of Spinoza "meaningless chewing around".
  - muninn_ 9y ago
    I think it's sad that your view that philosophy is worthless has become more of a mainstream feeling than it ought to be. Not everything needs to be measured in terms of usefulness. Interestingly enough, I think you find many developers have a natural affinity for philosophy. All this talk about AI, for example, is just philosophy.
  - pcmaffey 9y ago
    Most philosophers throughout history would laugh at their body of knowledge being measured in terms of utility. And then would become saddened that economics has become the dominant global value system.
    - notduncansmith 9y ago
      Economics is the word for the dominant global value system. I think a sad philosopher has more work to do.
  - pjmorris 9y ago
    I'd argue that philosophy may become more important than ever as we contemplate the ethics of handing decisions like who's going to die in a trolley-problem type self-driving car accident over to machines.
    - maxerickson 9y ago
      It'll be easier to (metaphorically) untie people from the tracks. Program slow speed limits in dangerous areas, install sensor augmentation in areas where vehicle located sensors don't work well, rebuild roads to work better, etc.
  - Ixiaus 9y ago
    A bit parochial to say that Philosophy--as it is taught or studied today--is useless, in my opinion. You're entitled to that thought but you might be surprised by the utility Philosophy has to offer. I think Philosophy studied with other disciplines is a particularly potent combination. Of my friends who have done this (they are very few) they are perhaps the most inspiring and trenchant thinkers I know.
- Jtsummers 9y ago
  No one would bring you a bucket of negative apples. Positive and negative ought to be thought of as directions, while the absolute value is the magnitude. A bucket of apples given to you has a positive direction (increasing your supply). A bucket of apples being taken from you has a negative direction (decreasing your supply).
  - KGIII 9y ago
    Of course you can't bring me a bucket of negative apples. That's the point. It very much is a philosophy and doesn't always relate to reality. Reality is, you can never really have a bucket of negative apples.
    - vinchuco 9y ago
      But it's your fault no one can bring you this bucket: You want 'negative apple' entities but ask for (what is interpreted as) a negative number of apple entities. A key distinction here is that one has a defined implementation. Making that distinction is philosophy. Talking within the framework of implementation is mathematics.
      - KGIII 9y ago
        Right. It's very much a philosophy. And it's not really my fault that nobody can bring me negative apples. It's not like I made the rules! Sheesh! ;-)
        vinchuco 9y ago
        You did! You posed the question! I just don't see the value in conflating both terms (philos and math) when there's a distinction pointed out above.
        KGIII 9y ago
        No, I didn't make the rules. Reality is what reality is. They don't let me decide the rules for reality. That's probably for the best. You can't bring me a bucket of negative apples. You can bring representations of future apples, or something like that. However, negative apples do not actually exist. In all these years, nobody has brought me a basket of negative apples.
- smohare 9y ago
  As another fellow mathematician (albeit pure), I take some issue with your argument. Rather, I'd say the collection of apples is a clear manifestation of a monoid that is not a group. Why should all physical systems form a group? It seems almost intellectually dishonest to suggest that a particular non-example is evidence against the universality of a mathematical object. Are symmetry groups not "physical" enough? I can rotate a piece of paper in the plane and then undo this operation just fine. Algebraic and topological structures seem to manifest themselves readily within nature. I should think infinities provide greater concern.
  - deleted 9y ago
    [deleted]
  - KGIII 9y ago
    The post may not have been clear enough, but the point is that it's impossible to really have negative apples. You can owe apples, but you can't physically possess them. Namely, pointing out how it remains a philosophical pursuit. As for infinities... I'm not sure if they drove Cantor insane or if he was already insane. Some infinities are bigger than others, and I have strong opinions about infinity. 0.999... = 1 Horrible! ;-)
    - BoiledCabbage 9y ago
      I disagree, I can physically possess 3 apples no more nor less than I can physically possess -3 apples. To say "I have three apples" is merely a way of modeling a configuration of the world. The apples may be in my pocket, or in my hand, or in a bag I'm holding, or even on my desk in front of me. They may be in the barn out back, but in all cases we can say I have 3 apples. Saying "i have 3 apples" is a simplification of a ton of configurations of the state of the world where I have ownership of apples. Similarly I have -3 apples is a summary of all the configurations of the world where I owe three apples. Neither is any more real than the other. And yes it is philosophical.
  - smnplk 9y ago
    >> Algebraic and topological structures seem to manifest themselves readily within nature. I should think infinities provide greater concern. So basically what you are saying is that, we can't be so certain that we humans invented math. Math is there, structures were there to be discovered ? Why is there such a thing as monster group ? We named those things, but if we didn't exists, it would still be there in the abstract world ? This is what is bugging me lately, I can't decide if math was invented or not. Some parts may be invented, but some parts may be discovered.
- SilasX 9y ago
  Not sure what that remark was in reference to, but ... yes, they can bring you negative apples. Your enterprise can have an obligation to deliver apples to someone, which has all the same effects as negative apples and should be booked as such. If you acquired an enterprise with such an obligation, they gave you negative apples. It's important to account for them that way in e.g. seeing the implications of the Put-Call Parity Theroem.
  - KGIII 9y ago
    See, there's no such thing as negative apples. Sure, they exist in math - but they don't exist in reality. You can possess a chit that says you owe apples but, in reality, you only have said chit and do not actually possess negative apples. The post may not have been clear enough, but the point was to show that it's still very much a philosophy.
- smegel 9y ago
  > nobody has successfully brought me a bucket of negative apples. But when they did I am going to take the square root of the number of them and break the universe.
  - vinchuco 9y ago
    I should have thought of this sooner. Since they're ALREADY talking about purely imaginary apples, take n of these. Arrange them in the bucket so as to make a square of side length n (of said imaginary apples). The result is n^2 negative apples and a really bad math joke.
- goodbye- 9y ago
  It must be extremely difficult for you to drive around in your car that is unable to go in reverse. Math is fascinating. It is also just a tool.
vijayr 9y ago
For someone who has forgotten math (other than high school level) and never did philosophy, what books/courses are good to start with (especially philosophy)?
- subwayclub 9y ago
  The community college course I took past spring used Thomas Nagel (What Does It All Mean?) supplemented with some handouts. I would recommend a class because part of the doing of philosophy is the discussion. Plus the classroom is one of the few places where you can delve into the nuances of an issue without being considered pedantic or boorish.

swayvil 9y ago

It's like being really good at a video game that only you and maybe 2 other guys play. You talk about the dungeons, monsters, level-bosses, etc but nobody can relate.

deleted 9y ago
[deleted]

paultopia 9y ago

Interestingly, this reads like the experience of expert knowledge in almost anything. Like, I have expert knowledge in law, and I'm working toward at least more advanced (though far far from expert) knowledge in coding-things, and it very nicely matches how I interact with legal problems and can dimly see glimmers off at the end of a long tunnel in coding problems. But at the same time, it's inspiring as all hell, because mathematics from the outside doesn't seem like the kind of thing that could ever produce the same satisfying and effective experiences of cognition.

nopinsight 9y ago

I somehow feel like this is written by Terence Tao, based on the quality, style, and length that are similar to what's on his blog. Anyone else shares the feeling or has a different guess? Edit: In a later part, the answer says "Terence Tao is very eloquent about this here: ..." so likely not (based on factors like typical personality of a mathematician, but I could be wrong since I don't know him personally).

deleted 9y ago
[deleted]
vinchuco 9y ago
https://en.wikipedia.org/wiki/Stylometry https://en.wikipedia.org/wiki/Stylometry
HiroshiSan 9y ago
that later part could be a nice red herring for those who think it is Terry.

jzl 9y ago

There are many wonderful passages in the book "Genius" by James Gleick that describe Richard Feynman's deep intuitive understanding of calculus. He also talks about it a bit in his own works such as Surely You're Joking Mr. Feynman. Highly recommend if you want to see lots of interesting answers to this question. His key approach was that he always wanted to understand things based on "first principles". So he would usually take on a new mathematical frameworks by breaking them down and re-deriving them on his own from scratch. Reminds me of how Ben Franklin taught himself to write by reading famous passages and then attempting to write them from memory, comparing his version to the original version afterward and seeing what he did differently and why.