Notes on "Love and Math" by Edward Frankel
Book is called: Love and Math The Heart of Hidden Reality, Frankel's journey of love to discover the "mysterious underlying structure" of all mathematics through exploration and understanding of the Langlands Program
My questions: Is mathematics something separate from the physical and mental world, as Frankel suggests in his discussion of symmetries? (p. 23). He says this is the Platonic World of mathematics. But I just don't get what the world would be? What world would be outside the physical and mental world? Suggests that there is "another" world that we don't have (immediate) access to in which math is produced? So there is a "square" out there of which we slowly learn the properties through testing theories? And squares "here" are just imitations of that square?
Why does it follow that if some mathematical principle (like symmetries) is "objective" that there is a platonic world of math? Do all "objective" things reside in the "other " world? How is this "other" world (let's call it Pworld for platonic world), How is the P world different from the unknown physical or mental world?
Is the (potential) multiple universes in the pworld?
Chapter 2: through math, the "smallest" particles were discovered (quarks).
I "get" this conclusion but the rest of the chapter made no sense to me. When he talks about the circle and the square and that they have symmetry because you can rotate them and at certain points (well, for a circle all points) they look exactly like they did before. But not sure how this fits with a butterfly (or a human) because you rotate it and it doesn't ever look like it did when you started except when you start, so how is that an example of having any symmetry? And, why/how is this related to math? Frankel states, "Mathematics is about the study of such abstract objects and concepts" (p. 21). Things that belong to the same "group" of symmetry (the circle group, for example) have the same properties, is independent of interpretation and will not change over time (this seems a guess though since all time has not passed. . . ). These properties of symmetry, I guess, is a mathematical concept . . .
Chapter 3: personal narrative of Frankel's failure because of his Jewish background to get admission to Moscow U where he wanted to take applied math
Chapter 4: He gets into the Institute of Oil and Gas because they basically accept all the rejected Jewish geniuses.
Chapter 5: Frankel gets assigned a problem from a noted mathematician that has to do with Braids. In this chapter, he explains Braids as he explains the problem he was presented with. I think I got the general idea of braids in that they are "string" attached
to a transparent board that can intersect/cross but can't become disconnected from the boards nor can it become entangled with itself. Braids with "n" threads form one "group" We can also number the overlaps of a braid within a group (Frankel states that the "braid is completely determined by the number of overlaps" (p. 52), but in this case I'm not quite sure what determined means). However, what this has to do with anything is beyond my understanding (he gives a couple of practical applications but I still didn't understand how the braids were useful).
He states that braids are another "group" that does NOT have to do with symmetries, which I guess introduces the notion that groups are important and have many diverse characteristics. I guess what makes them "groups" is that all the "things" in the group have the same properties . . .
I think he might be saying that these math concepts describe certain groups and then can be useful in other applications. But by themselves, these braids are simply theoretical math concepts (I think . . .)
Chapter 6: Discussion of the way that Gelfand ran his math seminars, suggesting that he was a hard task master but that his expectations met the abilities of those he worked with.
Chapter 7: Introduction to Langlands Grand Theory. Langlands wanted to bring together all mathematical cultures into one theoretical paradigm, and argues Frankel, this paradigm was organized around symmetries. Frankel explains how we can incorporate irrational numbers into the rational number system. This new system that we develop is enhanced because it has symmetries. "By a symmetry", writes Frankel, "I mean here a rule that assigns a new number to whatever number we begin with" (p. 74). He adds, "a given symmetry transforms each number to another number from the same numerical system". these symmetries from the "Galois group", p. 75. Galois took polynomial equations and suggested we focus on the group of symmetries of the number field obtained by joining the solutions to the equations to the rational numbers (the Galois group) (p. 77).. From this Galois was able to tell which solutions were actually solvable. Langlands tied this Galois group to Harmonics.
Chapter 8 (Okay, I almost gave up here). Beginning of this chapter is a bunch of formulas but the point being made is that equations that appear really complex can be solved through these hidden Langlands functions that take seemingly chaotic information and bring order to it. While I can see some of the order in the equations I don't see it as "mysterious" like Frankel does (p. 91). It seems like it's the opposite of mysterious, like it is revealing something that we didn't know before but was always there (and of course it was always there because it wasn't always there then we'd have to say somehow we "invented" it or something dubious like that. Isn't everything already here?). In fact, it seems to be de-masking the mysterious. These Langland functions also bring together seemingly disparate areas of math, in the example here (that is beyond my understanding) it brings together numbers theory and harmonic analysis.
Chapter 9: Frankel states that the Langlands program points to some universal phenomena that connects across different fields of mathematics (possible a Grand Unified Theory), p. 97. This chapter appears to be showing how Number theory, curves over finite fields and Reimann surfaces (three columns) are parts of the same big picture or are analogous to one another. And if number theory is similar to harmonic analysis, then in some way it too has to be related.
Chapter 10:
[Are all tools of the mind based on mathematics? Is mathematics required to understand the abstract. Clocks use numbers to represent time (or whatever time is), maps use numbers to represent space, the internet is built on numbers to send, code and decode information (data). Could we "see" an abstract world without mathematics of some kind? Math comes to us in our attempt to make sense of what fills "space" and "space" itself. When we don't understand what is "out there", we have to find some mathematical way of conceptualize the abstract in a formula. Analogies are another way we attempt to conceptualize that which is confusing . . .Is this what Hofstader addresses in Surfaces and Essences?]
Chapter 11-13 Frankel gets to Harvard
Chapter 14
In this chapter we learn of sheaves. But to get there, first we learn of "vector spaces" which is where numbers have been replaced by vector spaces (so a 1 is a line, so 1 + 1 is adding of those lines, 2 is a plane, etc), [seems like a way to make numbers go from abstract to concrete]. Vector spaces are richer and much more complex than numbers. Vector spaces form a "category", whereas numbers form sets. In the category, we can focus on how vectors interact with one another. When numbers are replaced by vector spaces, each function becomes a rule that assigns to the points in a manifold a vector space. The rule is called a sheaf. For functions, numbers are assigned to points; for sheaves vectors are assigned. Sheaves are categorifications of functions just like vectors are categorifications of numbers.
Sheaves can replace functions in the second of the three columns discussed in chapt 9 (number theory, curves over finite fields and Reimann surfaces) and thus make the middle and right columns analogous. The Galois group and automorphic functions do not have a direct relation to Reimann surfaces. Through the Grothendieck dictionary, we can replace the automorphic with sheaves that satisfy properties of the automorphoc functions and we call these automorphic sheaves.
Our three columns (helping build the grand theory that the Langlands program allows us) now look like
Column 1
number theory
galois group
automprohic functions
Column 2 (my stupid head program won't let me make these columns next to one another as they should be)
curves over finite fields
galois group
automorphic functions or automorphic sheaves
Column 3
Riemann surfaces
fundamental group
automoprhic sheaves
Chapter 15 Frankel summarizes his dissertation which has something to do with Lie groups and Langland functions but I couldn't understand any of it except that it is part of the effort to develop a unity between geometry, algebra and arithmetic.
Chapter 16
Fermions--building blocks of matter like electrons, quarks that have 1/2 integer spin (cannot be in the same space at the same time)
Bosons--particles that carry forces like photons, the Higgs (can be in the same space at the same time)
Bosons and Fermions can be exchanged and have symmetry according to a theory of super symmetry but so far this hasn't been found in nature.
We do not know if quantum electromagnetic duality exists in the real world but in the super symmetric theory it does
Says in this chapter, "Mathematical truths seem to exist objectively and independently of the both the physical world and the human brain" (p. 202). What does that even mean?
Seems to be saying that symmetry is a way of making really confusing things easier to work with . . . even if the symmetries aren't entirely mapped onto the real world. . .
Chapter 17
So, if I were to summarize this chapter I would just have to say that they added quantum physics as a fourth column in the above Langlands columns.
Chapter 18
I get when he says that mathematics is separate from the "human world", that is we did not create math or discover it. It's always present. But isn't it the physical world, not separate from the physical world? He argues math is outside the physical world because there are mathematical concepts that do not refer to any known concept. But that doesn't mean they aren't in the physical world. They just aren't in the known physical world
[but could there be (isn't there) a formula for love in the sense that the physical world is already present, what is going to happen is going to happen and some what is going to happen has been labeled "love" by us. If those something are the same, then they would have a formula, but really what is love?]
My questions: Is mathematics something separate from the physical and mental world, as Frankel suggests in his discussion of symmetries? (p. 23). He says this is the Platonic World of mathematics. But I just don't get what the world would be? What world would be outside the physical and mental world? Suggests that there is "another" world that we don't have (immediate) access to in which math is produced? So there is a "square" out there of which we slowly learn the properties through testing theories? And squares "here" are just imitations of that square?
Why does it follow that if some mathematical principle (like symmetries) is "objective" that there is a platonic world of math? Do all "objective" things reside in the "other " world? How is this "other" world (let's call it Pworld for platonic world), How is the P world different from the unknown physical or mental world?
Is the (potential) multiple universes in the pworld?
Chapter 2: through math, the "smallest" particles were discovered (quarks).
I "get" this conclusion but the rest of the chapter made no sense to me. When he talks about the circle and the square and that they have symmetry because you can rotate them and at certain points (well, for a circle all points) they look exactly like they did before. But not sure how this fits with a butterfly (or a human) because you rotate it and it doesn't ever look like it did when you started except when you start, so how is that an example of having any symmetry? And, why/how is this related to math? Frankel states, "Mathematics is about the study of such abstract objects and concepts" (p. 21). Things that belong to the same "group" of symmetry (the circle group, for example) have the same properties, is independent of interpretation and will not change over time (this seems a guess though since all time has not passed. . . ). These properties of symmetry, I guess, is a mathematical concept . . .
Chapter 3: personal narrative of Frankel's failure because of his Jewish background to get admission to Moscow U where he wanted to take applied math
Chapter 4: He gets into the Institute of Oil and Gas because they basically accept all the rejected Jewish geniuses.
Chapter 5: Frankel gets assigned a problem from a noted mathematician that has to do with Braids. In this chapter, he explains Braids as he explains the problem he was presented with. I think I got the general idea of braids in that they are "string" attached
to a transparent board that can intersect/cross but can't become disconnected from the boards nor can it become entangled with itself. Braids with "n" threads form one "group" We can also number the overlaps of a braid within a group (Frankel states that the "braid is completely determined by the number of overlaps" (p. 52), but in this case I'm not quite sure what determined means). However, what this has to do with anything is beyond my understanding (he gives a couple of practical applications but I still didn't understand how the braids were useful).
He states that braids are another "group" that does NOT have to do with symmetries, which I guess introduces the notion that groups are important and have many diverse characteristics. I guess what makes them "groups" is that all the "things" in the group have the same properties . . .
I think he might be saying that these math concepts describe certain groups and then can be useful in other applications. But by themselves, these braids are simply theoretical math concepts (I think . . .)
Chapter 6: Discussion of the way that Gelfand ran his math seminars, suggesting that he was a hard task master but that his expectations met the abilities of those he worked with.
Chapter 7: Introduction to Langlands Grand Theory. Langlands wanted to bring together all mathematical cultures into one theoretical paradigm, and argues Frankel, this paradigm was organized around symmetries. Frankel explains how we can incorporate irrational numbers into the rational number system. This new system that we develop is enhanced because it has symmetries. "By a symmetry", writes Frankel, "I mean here a rule that assigns a new number to whatever number we begin with" (p. 74). He adds, "a given symmetry transforms each number to another number from the same numerical system". these symmetries from the "Galois group", p. 75. Galois took polynomial equations and suggested we focus on the group of symmetries of the number field obtained by joining the solutions to the equations to the rational numbers (the Galois group) (p. 77).. From this Galois was able to tell which solutions were actually solvable. Langlands tied this Galois group to Harmonics.
Chapter 8 (Okay, I almost gave up here). Beginning of this chapter is a bunch of formulas but the point being made is that equations that appear really complex can be solved through these hidden Langlands functions that take seemingly chaotic information and bring order to it. While I can see some of the order in the equations I don't see it as "mysterious" like Frankel does (p. 91). It seems like it's the opposite of mysterious, like it is revealing something that we didn't know before but was always there (and of course it was always there because it wasn't always there then we'd have to say somehow we "invented" it or something dubious like that. Isn't everything already here?). In fact, it seems to be de-masking the mysterious. These Langland functions also bring together seemingly disparate areas of math, in the example here (that is beyond my understanding) it brings together numbers theory and harmonic analysis.
Chapter 9: Frankel states that the Langlands program points to some universal phenomena that connects across different fields of mathematics (possible a Grand Unified Theory), p. 97. This chapter appears to be showing how Number theory, curves over finite fields and Reimann surfaces (three columns) are parts of the same big picture or are analogous to one another. And if number theory is similar to harmonic analysis, then in some way it too has to be related.
Chapter 10:
[Are all tools of the mind based on mathematics? Is mathematics required to understand the abstract. Clocks use numbers to represent time (or whatever time is), maps use numbers to represent space, the internet is built on numbers to send, code and decode information (data). Could we "see" an abstract world without mathematics of some kind? Math comes to us in our attempt to make sense of what fills "space" and "space" itself. When we don't understand what is "out there", we have to find some mathematical way of conceptualize the abstract in a formula. Analogies are another way we attempt to conceptualize that which is confusing . . .Is this what Hofstader addresses in Surfaces and Essences?]
Chapter 11-13 Frankel gets to Harvard
Chapter 14
In this chapter we learn of sheaves. But to get there, first we learn of "vector spaces" which is where numbers have been replaced by vector spaces (so a 1 is a line, so 1 + 1 is adding of those lines, 2 is a plane, etc), [seems like a way to make numbers go from abstract to concrete]. Vector spaces are richer and much more complex than numbers. Vector spaces form a "category", whereas numbers form sets. In the category, we can focus on how vectors interact with one another. When numbers are replaced by vector spaces, each function becomes a rule that assigns to the points in a manifold a vector space. The rule is called a sheaf. For functions, numbers are assigned to points; for sheaves vectors are assigned. Sheaves are categorifications of functions just like vectors are categorifications of numbers.
Sheaves can replace functions in the second of the three columns discussed in chapt 9 (number theory, curves over finite fields and Reimann surfaces) and thus make the middle and right columns analogous. The Galois group and automorphic functions do not have a direct relation to Reimann surfaces. Through the Grothendieck dictionary, we can replace the automorphic with sheaves that satisfy properties of the automorphoc functions and we call these automorphic sheaves.
Our three columns (helping build the grand theory that the Langlands program allows us) now look like
Column 1
number theory
galois group
automprohic functions
Column 2 (my stupid head program won't let me make these columns next to one another as they should be)
curves over finite fields
galois group
automorphic functions or automorphic sheaves
Column 3
Riemann surfaces
fundamental group
automoprhic sheaves
Chapter 15 Frankel summarizes his dissertation which has something to do with Lie groups and Langland functions but I couldn't understand any of it except that it is part of the effort to develop a unity between geometry, algebra and arithmetic.
Chapter 16
Fermions--building blocks of matter like electrons, quarks that have 1/2 integer spin (cannot be in the same space at the same time)
Bosons--particles that carry forces like photons, the Higgs (can be in the same space at the same time)
Bosons and Fermions can be exchanged and have symmetry according to a theory of super symmetry but so far this hasn't been found in nature.
We do not know if quantum electromagnetic duality exists in the real world but in the super symmetric theory it does
Says in this chapter, "Mathematical truths seem to exist objectively and independently of the both the physical world and the human brain" (p. 202). What does that even mean?
Seems to be saying that symmetry is a way of making really confusing things easier to work with . . . even if the symmetries aren't entirely mapped onto the real world. . .
Chapter 17
So, if I were to summarize this chapter I would just have to say that they added quantum physics as a fourth column in the above Langlands columns.
Chapter 18
I get when he says that mathematics is separate from the "human world", that is we did not create math or discover it. It's always present. But isn't it the physical world, not separate from the physical world? He argues math is outside the physical world because there are mathematical concepts that do not refer to any known concept. But that doesn't mean they aren't in the physical world. They just aren't in the known physical world
[but could there be (isn't there) a formula for love in the sense that the physical world is already present, what is going to happen is going to happen and some what is going to happen has been labeled "love" by us. If those something are the same, then they would have a formula, but really what is love?]
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