8 ms·

Mathematics as thought

147 points by magoghm 8y ago

allemagne 8y ago
I thought this was a really beautiful article. All I have to add is that it seemed to avoid discussing philosophy as a discipline, even when it seems impossible to ignore. For example: >Perhaps more than any other subject, mathematics is about the study of ideas.
astroalex 8y ago
> For instance, say you’re in a race at school. You do surprisingly well and beat most of your classmates. All things being equal, the next time around, you’re actually not likely to do as well, relative to the other runners. If the events are independent (as I believe the author is assuming at this point), then your performance in the first race has no effect on your performance in the next race, right? Maybe it's just a poor choice of wording, but it comes dangerously close to reinforcing a common misconception about probability (Gambler's fallacy).
- CamTin 8y ago
  If you have a competition where everyone flips coins, and you are eliminated as soon as you flip a tail, and you win, it's likely that you will also win the next time. This is not because the events are not independent, but because you are always unlikely to win. Likewise, if you get a royal flush in a hand of poker, you probably won't get as good a hand next time. Again, this is not because the deals aren't independent events, but because it is always unlikely to get a royal flush.
  - astroalex 8y ago
    Good points. > Likewise, if you get a royal flush in a hand of poker, you probably won't get as good a hand next time. I understand what you mean, but I think that most people who haven't taken a probability class would misinterpret this. A sentence of the form "if [this] then [that]" is usually meant to express dependence. Don't you think that most people would interpret your sentence as: "Because I got a royal flush this time, I probably won't get one next time."? I think the average pop-sci article reader will get confused by this.
    - leetcrew 8y ago
      it's really hard to map subtle statements about probability to natural English sentences. pretty much any non-awkward formulation of that sentence is going to admit misinterpretations.
  - antidesitter 8y ago
    > it's likely that you will also win the next time. This is... because you are always unlikely to win. Did you mean to write “it’s not likely that you will also win next time”?
    - CamTin 8y ago
      I did. Unfortunately the editing window is closed. Thank you.
- buddhistan 8y ago
  I don't think it's phrased to correlate the second outcome dependent on the first. That part of the article was referring to the racer regressing to their mean. With the assumption that they overperform in their first race based on the phrasing "surprisingly well", then the conclusion is still correct because the likelihood of them performing to their average performance level (whatever that may be) is still more likely than overperforming.
- Someone 8y ago
  I don’t see why you think that. Let’s say each of your classmates has a number of dice representing their innate ability, and rolls all of them. You have 5 dice, do surprisingly wel and roll four sixes and a four for a total of 28, beating some classmates who have 6, 7, or even 8 dice. If you repeat this, it’s unlikely you will do similarly well.
- btilly 8y ago
  No, the author is not assuming that. And in fact the opposite is true. The assumption is that your performance on any given race is some combination of luck and ability. If you perform extremely well, it should be assumed that both were in your favor that time. The next time you'll still have whatever ability is in play, but you are unlikely to have luck. The result is that your performance is likely to be good, though not stellar. You racing one day or the next are not independent events - your ability creates a correlation. But your performance one day is also not a linear prediction of your expected performance the next. This is called regression to the mean. It comes into play in everything from stock picking to poker players.
  - Jtsummers 8y ago
    To diagram it out: a = ability l = luck p = performance a l \ / \ / v p Each (ability and luck) has a contributing factor. Your ability is more likely to be consistent (within some stretch of time). Your luck is less consistent. So if you average a 7 minute mile, and you somehow run it in 6, unless your fundamental ability has changed it was luck (weather, mental state, wear on your shoes) that pushed you so far outside your norm, and you should anticipate returning to your average in the future.
    - antidesitter 8y ago
      What is “luck” supposed to mean in your diagram?
      - Jtsummers 8y ago
        Any contributing factor outside your base ability. Weather (too hot, too cold, just right), mental state (did someone put you on edge, sleep poorly), etc. So some of those are more or less under your control. Maybe you can control your mental state better than me. Me, a particularly stressful day at work doesn't lend to the quiet mind I need for a run. Those are my 36 minute 5km days. But nicer weather, seeing the geese on the pond as they migrate, can put my mind at ease even on those stressful days and I might pull off one of my faster runs. I can't control that.
- nimonian 8y ago
  I was surprised at this passage, too. It begins by quoting the Oxford dictionary, which always raises alarm bells that the author's ideas on the topic are inchoate, but the wording of the 'racing' example is an unforgivable blunder. The 'mean' referred to in 'regression to the mean' is the mean of the individual, not of the pack.
Meai 8y ago
"How mathematics works rests on no absolute timeless standard, despite what many assume today, given its precision and efficacy" Exactly this is key to teach people to like math, stop trying to train me in it. It's like math has become so saddled with politics or ego that people want to train everybody in the same system of values and they are willing to lie to get you to use to the same symbols, the same way of thinking, etc. Disclaimer, I'm really bad at math. But I hate how math and physics are taught, it should always be taught from a spirit of exploration as if you are discovering new islands or continents and you get to name the things you discover. Try getting a math professor to admit that he doesn't actually have an intuitive understanding for any of the stuff he teaches. I believe the vast majority probably don't have any such understanding but they are understandably scared to admit it. It's a cycle of lies. Who are you to say that I have to use that particular symbol or term to describe this math? People have a large stake in interopability between mathematicians so they teach us these symbols as if they are facts of nature when in reality they are just arbitrary drawings that people invented to name stuff that they found in nature. Fields of math stack on top of each other because they happen to work together but in reality we dont know why. They just do and it's interesting that they do so we keep doing it that way because it works and hasnt broken yet and it's useful. There were cultures that did math with a completely different set of symbols, not just symbols but even way of thinking. Multiplication, addition, entire base ideas were thought of differently. Yet our current way of doing math is taught to us as dogmatic, accept it or you are a troublemaker. No wonder people dont like math, nobody wants to be a slave to somebody else's ideas and value system. Every math class should start with something like "This is the symbol for addition: '+'. This was invented by some person, you could use something else, you could try to find some other way to add numbers but the way we teach you to think has proven to be fairly fast and convenient so we teach it." That in my opinion is how you make people interested in math because you treat them as equals instead of subjects to be trained in your favorite cultural way of doing math. Math should be seen like a weird natural phenomenon that we observe, from the getgo. From childhood on. Explain it like that and I believe kids (and adults) will love to discover more about it.
- thrmsforbfast 8y ago
  This strikes me as fairly unreasonable. The full range of human thought is probably possible to express in the language my sibling and I made up as children, but I'm not arrogant enough to assert that my teachers are all incompetent liars just because they refuse to grade essays written in our made-up language. Every mathematics professor I've ever known would readily agree that part of teaching mathematics is teaching how to communicate ideas using the established linguistic customs of modern mathematics. That's reasonable, just like it's reasonable that literature professors insist on essays written in modern English.
blt 8y ago
Opening up with the claim that computer software is "based on the ideas of Claude Shannon" is strange. Does anyone seriously believe that Shannon's contributions to the foundations of computing stand so far above others'?
- danharaj 8y ago
  I do, but that sentence is weird.
- TheOtherHobbes 8y ago
  Turing - On Computable Numbers, with an Application to the Entscheidungsproblem - 1937 Shannon - A Mathematical Theory of Communication - 1948 Shannon calculated the entropy of arbitrary symbol systems over noisy communication channels, and made some first steps towards practical data compression algorithms. Turning formalised the concept of computability using an abstraction of a mechanised logic system. I'd say Turing wins easily - we still say logical systems are Turing Complete rather than Shannon Compatible.
  - mbrock 8y ago
    Don’t forget Shannon’s 1937 master’s thesis, A Symbolic Analysis of Relay and Switching Circuits.
    - yesenadam 8y ago
      Yeah, he actually had the idea that you could use the 19th C Boolean logic and switching circuits to do mathematical operations, making modern computers possible. It's disappointing that he's not better known for this, to say the least. e.g. that anyone on here doesn't know his huge importance.
- jonsen 8y ago
  Yes, his masters thesis in 1937: https://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuits https://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_a...
- 8y ago
hodgesrm 8y ago
I was a bit disappointed that this article did not offer more practical examples of the ties between the development of mathematics and the history of ideas in general. For instance, the vision of mathematicians like Hilbert and Russell of a cohesive mathematics defined from first principles seems a very Victorian notion that the universe is knowable if you simply search hard enough. It's the same enlightment frame of mind that resulted in intellectual creations like the Oxford English Dictionary. Similarly the results of Gödel, Church, Turing, and others showing the limits of mathematical consistency seem of a piece with the confounding discoveries of Quantum physics, which replaced the static models of classical physics. They seem to correspond with a darker vision of the limitations of human intellect that emerge in abstract art and poetry like Eliot's 'The Wasteland' around the same time. If you push this too far you end up sounding like an idiot (perhaps this is already there) but there are clearly discernible connections across science, art, literature, and politics both now and deep into the past. It might be a personal projection but Brahms concertos always strike me as the music of a people who believed in an orderly, Newtonian universe and upheld the static political order of the Ancien Regime. At the end of the day most intellectuals read the same books and attend the same salons, however defined. There is continuous cross-fertilization between different realms of thought. The resulting connections are there if you just look for them.
- pegasus 8y ago
  Related book: https://www.amazon.com/Mathematics-Roots-Postmodern-Thought-Vladimir/dp/0195139674 https://www.amazon.com/Mathematics-Roots-Postmodern-Thought-...
- hodgesrm 8y ago
  Edit: 'Brahms' should be 'J.S. Bach'.
bmc7505 8y ago
Nice parallel with Iverson's 1979 Turing Award Lecture, "Notation as a Tool for Thought": http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pdf http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pd...

147 points by magoghm 8y ago

I thought this was a really beautiful article. All I have to add is that it seemed to avoid discussing philosophy as a discipline, even when it seems impossible to ignore. For example: >Perhaps more than any other subject, mathematics is about the study of ideas.

astroalex 8y ago

> For instance, say you’re in a race at school. You do surprisingly well and beat most of your classmates. All things being equal, the next time around, you’re actually not likely to do as well, relative to the other runners. If the events are independent (as I believe the author is assuming at this point), then your performance in the first race has no effect on your performance in the next race, right? Maybe it's just a poor choice of wording, but it comes dangerously close to reinforcing a common misconception about probability (Gambler's fallacy).

CamTin 8y ago
If you have a competition where everyone flips coins, and you are eliminated as soon as you flip a tail, and you win, it's likely that you will also win the next time. This is not because the events are not independent, but because you are always unlikely to win. Likewise, if you get a royal flush in a hand of poker, you probably won't get as good a hand next time. Again, this is not because the deals aren't independent events, but because it is always unlikely to get a royal flush.
- astroalex 8y ago
  Good points. > Likewise, if you get a royal flush in a hand of poker, you probably won't get as good a hand next time. I understand what you mean, but I think that most people who haven't taken a probability class would misinterpret this. A sentence of the form "if [this] then [that]" is usually meant to express dependence. Don't you think that most people would interpret your sentence as: "Because I got a royal flush this time, I probably won't get one next time."? I think the average pop-sci article reader will get confused by this.
  - leetcrew 8y ago
    it's really hard to map subtle statements about probability to natural English sentences. pretty much any non-awkward formulation of that sentence is going to admit misinterpretations.
- antidesitter 8y ago
  > it's likely that you will also win the next time. This is... because you are always unlikely to win. Did you mean to write “it’s not likely that you will also win next time”?
  - CamTin 8y ago
    I did. Unfortunately the editing window is closed. Thank you.
buddhistan 8y ago
I don't think it's phrased to correlate the second outcome dependent on the first. That part of the article was referring to the racer regressing to their mean. With the assumption that they overperform in their first race based on the phrasing "surprisingly well", then the conclusion is still correct because the likelihood of them performing to their average performance level (whatever that may be) is still more likely than overperforming.
Someone 8y ago
I don’t see why you think that. Let’s say each of your classmates has a number of dice representing their innate ability, and rolls all of them. You have 5 dice, do surprisingly wel and roll four sixes and a four for a total of 28, beating some classmates who have 6, 7, or even 8 dice. If you repeat this, it’s unlikely you will do similarly well.
btilly 8y ago
No, the author is not assuming that. And in fact the opposite is true. The assumption is that your performance on any given race is some combination of luck and ability. If you perform extremely well, it should be assumed that both were in your favor that time. The next time you'll still have whatever ability is in play, but you are unlikely to have luck. The result is that your performance is likely to be good, though not stellar. You racing one day or the next are not independent events - your ability creates a correlation. But your performance one day is also not a linear prediction of your expected performance the next. This is called regression to the mean. It comes into play in everything from stock picking to poker players.
- Jtsummers 8y ago
  To diagram it out: a = ability l = luck p = performance a l \ / \ / v p Each (ability and luck) has a contributing factor. Your ability is more likely to be consistent (within some stretch of time). Your luck is less consistent. So if you average a 7 minute mile, and you somehow run it in 6, unless your fundamental ability has changed it was luck (weather, mental state, wear on your shoes) that pushed you so far outside your norm, and you should anticipate returning to your average in the future.
  - antidesitter 8y ago
    What is “luck” supposed to mean in your diagram?
    - Jtsummers 8y ago
      Any contributing factor outside your base ability. Weather (too hot, too cold, just right), mental state (did someone put you on edge, sleep poorly), etc. So some of those are more or less under your control. Maybe you can control your mental state better than me. Me, a particularly stressful day at work doesn't lend to the quiet mind I need for a run. Those are my 36 minute 5km days. But nicer weather, seeing the geese on the pond as they migrate, can put my mind at ease even on those stressful days and I might pull off one of my faster runs. I can't control that.
nimonian 8y ago
I was surprised at this passage, too. It begins by quoting the Oxford dictionary, which always raises alarm bells that the author's ideas on the topic are inchoate, but the wording of the 'racing' example is an unforgivable blunder. The 'mean' referred to in 'regression to the mean' is the mean of the individual, not of the pack.

Meai 8y ago

"How mathematics works rests on no absolute timeless standard, despite what many assume today, given its precision and efficacy" Exactly this is key to teach people to like math, stop trying to train me in it. It's like math has become so saddled with politics or ego that people want to train everybody in the same system of values and they are willing to lie to get you to use to the same symbols, the same way of thinking, etc. Disclaimer, I'm really bad at math. But I hate how math and physics are taught, it should always be taught from a spirit of exploration as if you are discovering new islands or continents and you get to name the things you discover. Try getting a math professor to admit that he doesn't actually have an intuitive understanding for any of the stuff he teaches. I believe the vast majority probably don't have any such understanding but they are understandably scared to admit it. It's a cycle of lies. Who are you to say that I have to use that particular symbol or term to describe this math? People have a large stake in interopability between mathematicians so they teach us these symbols as if they are facts of nature when in reality they are just arbitrary drawings that people invented to name stuff that they found in nature. Fields of math stack on top of each other because they happen to work together but in reality we dont know why. They just do and it's interesting that they do so we keep doing it that way because it works and hasnt broken yet and it's useful. There were cultures that did math with a completely different set of symbols, not just symbols but even way of thinking. Multiplication, addition, entire base ideas were thought of differently. Yet our current way of doing math is taught to us as dogmatic, accept it or you are a troublemaker. No wonder people dont like math, nobody wants to be a slave to somebody else's ideas and value system. Every math class should start with something like "This is the symbol for addition: '+'. This was invented by some person, you could use something else, you could try to find some other way to add numbers but the way we teach you to think has proven to be fairly fast and convenient so we teach it." That in my opinion is how you make people interested in math because you treat them as equals instead of subjects to be trained in your favorite cultural way of doing math. Math should be seen like a weird natural phenomenon that we observe, from the getgo. From childhood on. Explain it like that and I believe kids (and adults) will love to discover more about it.

thrmsforbfast 8y ago
This strikes me as fairly unreasonable. The full range of human thought is probably possible to express in the language my sibling and I made up as children, but I'm not arrogant enough to assert that my teachers are all incompetent liars just because they refuse to grade essays written in our made-up language. Every mathematics professor I've ever known would readily agree that part of teaching mathematics is teaching how to communicate ideas using the established linguistic customs of modern mathematics. That's reasonable, just like it's reasonable that literature professors insist on essays written in modern English.

blt 8y ago

Opening up with the claim that computer software is "based on the ideas of Claude Shannon" is strange. Does anyone seriously believe that Shannon's contributions to the foundations of computing stand so far above others'?

danharaj 8y ago
I do, but that sentence is weird.
TheOtherHobbes 8y ago
Turing - On Computable Numbers, with an Application to the Entscheidungsproblem - 1937 Shannon - A Mathematical Theory of Communication - 1948 Shannon calculated the entropy of arbitrary symbol systems over noisy communication channels, and made some first steps towards practical data compression algorithms. Turning formalised the concept of computability using an abstraction of a mechanised logic system. I'd say Turing wins easily - we still say logical systems are Turing Complete rather than Shannon Compatible.
- mbrock 8y ago
  Don’t forget Shannon’s 1937 master’s thesis, A Symbolic Analysis of Relay and Switching Circuits.
  - yesenadam 8y ago
    Yeah, he actually had the idea that you could use the 19th C Boolean logic and switching circuits to do mathematical operations, making modern computers possible. It's disappointing that he's not better known for this, to say the least. e.g. that anyone on here doesn't know his huge importance.
jonsen 8y ago
Yes, his masters thesis in 1937: https://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuits https://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_a...
8y ago

hodgesrm 8y ago

I was a bit disappointed that this article did not offer more practical examples of the ties between the development of mathematics and the history of ideas in general. For instance, the vision of mathematicians like Hilbert and Russell of a cohesive mathematics defined from first principles seems a very Victorian notion that the universe is knowable if you simply search hard enough. It's the same enlightment frame of mind that resulted in intellectual creations like the Oxford English Dictionary. Similarly the results of Gödel, Church, Turing, and others showing the limits of mathematical consistency seem of a piece with the confounding discoveries of Quantum physics, which replaced the static models of classical physics. They seem to correspond with a darker vision of the limitations of human intellect that emerge in abstract art and poetry like Eliot's 'The Wasteland' around the same time. If you push this too far you end up sounding like an idiot (perhaps this is already there) but there are clearly discernible connections across science, art, literature, and politics both now and deep into the past. It might be a personal projection but Brahms concertos always strike me as the music of a people who believed in an orderly, Newtonian universe and upheld the static political order of the Ancien Regime. At the end of the day most intellectuals read the same books and attend the same salons, however defined. There is continuous cross-fertilization between different realms of thought. The resulting connections are there if you just look for them.

pegasus 8y ago
Related book: https://www.amazon.com/Mathematics-Roots-Postmodern-Thought-Vladimir/dp/0195139674 https://www.amazon.com/Mathematics-Roots-Postmodern-Thought-...
hodgesrm 8y ago
Edit: 'Brahms' should be 'J.S. Bach'.

bmc7505 8y ago

Nice parallel with Iverson's 1979 Turing Award Lecture, "Notation as a Tool for Thought": http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pdf http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pd...