I have always found this a rather puzzling statement. What is this elusive "mathematical maturity" and how should one go about obtaining it? Let me return to the story from my previous post about misunderstanding a mathematical statement, which set me on this train of thought.
Whatever "mathematical maturity" might be, I know it is not something I have, but at least I'm starting to glimpse what it might look like and be like. I think one aspect - and perhaps the foundational one - is fluency in the language of mathematics, which includes being able to change how one interprets typography at will (and not being stuck on the idea that certain patterns of written squiggles mean only one thing all the time).
Someone fluent in linear algebra might look at this and think, "The absolute value of the inner product of two numbers x and y is less than or equal to the product of the norm of the first number x and the second number y."
However, when I see a theorem for the first time - as a most definitely not-mathematically mature student - I almost always have an initial moment of panic** where I think, I have no clue what this means!
After I calm down, my internal dialogue might go something like this:
To make it all worse, my brain keeps reverting back to high-school algebra mode where (x,y) is an ordered pair that gives a location within a coordinate plane. So my initial reading is by default wrong because this is a chapter on inner products***!
One reason I like Sheldon Axler's book Linear Algebra Done Right is simply because it's aimed at undergraduates and actually explains everything in words. Sometimes it is annoying that he doesn't use summation notation, but it's kind of nice as well that the notation isn't as compact because it is easier to read at a glance. I always end up writing out and expanding whatever Lax has so compactly and elegantly written, simply because he leaves lots of proofs to the reader - which is great if the reader actually knows how to prove things. Not only does he assume that the reader is familiar with much mathematical notation (which he often does not bother to define), but also basics of proof-writing. Again, it's great if you're at that level, but it is a huge struggle for me even to comprehend what he's saying.
Yet in comparing Axler's book to the far more advanced style of Lax, I can see that this maturity is not only fluency in reading and interpreting the language of mathematics, including understanding the contextual use of symbols and typography that would otherwise be ambiguous, but the ability to fill in the blanks and appreciate not only whether a proof is, well, proven (still where I'm at!) but also whether it's beautiful as well. Axler chooses to write his proofs in a way that can be unwieldy but for me is often easier to understand. Lax's tend to be short and slick, but do not necessarily yield insight into the meaning of a theorem.
This "maturity" also seems to be about one's ability to make the leap to abstraction, having the ability to reinterpret the meaning of symbols dynamically (vertical bar meaning absolute value, norm, etc.) and also generalize their meaning in more powerful ways. I think it can include having a visual or intuitive understanding of concepts and - even harder - being able to express those in mathematical language, and choosing a formalism and notation that is clear and "natural" for one's purposes.
In other words, I don't get stuck on the meaning of x. I am okay with it being a number, a vector, a matrix, or other strange object as long as all of those follow the rules of typographical manipulation that I am allowed to practice on that object. I can erase old definitions and use new definitions for the same symbols (the vertical bar) without constantly calling the old definition. I don't panic when I see unfamiliar notation because I have confidence that reading the definition will clear things up.
I guess all of this is still pretty basic stuff, but it's about as far as I can imagine as a beginning mathematics student! I am definitely at the stage of learning a new language and all the grammar and cultural conventions that come with it. I constantly have to think about what I'm doing, try to translate my thoughts into the appropriate notation, and hope that I'm saying what I think I'm saying. I aspire to the kind of fluency in which I speak the language of mathematics and can focus more on what I want to say and express than the mechanics of doing so. I think another handy thing would be able to have a better vocabulary to describe mathematical ideas and notation. Because I'm such a visual thinker, I often have diagrams, pictures, or even movies in my head, but struggle to communicate verbally what I'm imagining. (Fortunately, I can draw.)
* * *
But as to mathematical maturity? My professor (in the last post) had a gut-level emotional reaction when he saw my (badly-written) equation because - I presume - he is fluent enough in mathematics that he just knew it was formulated incorrectly, even without being able to articulate why. It just looked wrong.
I'm still worlds away from that. The only situation in which I have that kind of fluency is in English. As a native speaker, my grammar may not always be correct, but I usually know when something is wrong - without any recourse to the appropriate grammar rule. I know because I've read and heard enough that the the familiar pattern of sentence structure is deeply ingrained, so it stands out when something does not fit that pattern of words and phrases. Perhaps someday I'll be fluent enough in the logic of mathematics that I will be able to use pattern recognition rather than the laborious conscious application of new ideas to read and understand theorems.
To sum it all up, Larry Deneberg says it most eloquently. For him, mathematical maturity is
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* p. 79
(I rate my progress maturing mathematically by the length and intensity of this fear. It has declined from a solid minute of panic in the early days to a momentary blip because I know that if I dive in and read all the definitions, I will probably be able to understand what the statement means. Lesson 1 in mathematical maturity: Don't panic when seeing unfamiliar notation. After all, as a native English speaker, I don't panic when I see a word I don't know; I just look it up in a dictionary. But statements written in Russian seem pretty intimidating to me, as do many statements in mathematical language.)
***The inner product is also known as the scalar product or dot product.
Whatever "mathematical maturity" might be, I know it is not something I have, but at least I'm starting to glimpse what it might look like and be like. I think one aspect - and perhaps the foundational one - is fluency in the language of mathematics, which includes being able to change how one interprets typography at will (and not being stuck on the idea that certain patterns of written squiggles mean only one thing all the time).
* * *
For example, in the context of Peter Lax's Introduction to Linear Algebra and Its Applications or my current course, we might see a statement like this (the Schwartz inequality*):
| (x, y) | <= ||x|| ||y||
Someone fluent in linear algebra might look at this and think, "The absolute value of the inner product of two numbers x and y is less than or equal to the product of the norm of the first number x and the second number y."
However, when I see a theorem for the first time - as a most definitely not-mathematically mature student - I almost always have an initial moment of panic** where I think, I have no clue what this means!
After I calm down, my internal dialogue might go something like this:
Hmm. What do those vertical bars mean? Why am I taking the absolute value of coordinates (x,y)? That doesn't make sense. Are those bars supposed to mean the norm like they do in class? No, in this textbook the norm is written with a double bar || . || so the single vertical line must mean absolute value. Oh wait, the (x,y) doesn't mean the location on the plane, it's supposed to mean the inner product.
What's the inner product again? (flip back to definition) Okay, I'm supposed to take the sum of the product of all components of the vectors x and y. Hmm... what is a norm again? It's length, great - remember, norm means length. By definition it's the square root of the sum of the squares of the components of the vectors, but I could also think of it as the square root of the inner product of the vector with itself as ||x||2=(x,x)...And part of the problem here is that a lot of the notation is contextual. For example, vertical bars around a scalar (a number) mean absolute value: |k| or |9| . In class we drop the double bar || . || for norm so that if x is a vector, |x| means the norm of x, as we always take norms of vectors and not scalars.
To make it all worse, my brain keeps reverting back to high-school algebra mode where (x,y) is an ordered pair that gives a location within a coordinate plane. So my initial reading is by default wrong because this is a chapter on inner products***!
* * *
Returning to this idea of mathematical maturity, I have to say that one of my biggest frustrations with Lax's textbook is he assumes a much higher level of fluency in reading mathematics than I happen to possess, which makes it impossible for me to use the textbook as a self-contained learning experience. It was even harder last semester because it wasn't obvious to me where I could find the missing pieces or what words I would need to find them.One reason I like Sheldon Axler's book Linear Algebra Done Right is simply because it's aimed at undergraduates and actually explains everything in words. Sometimes it is annoying that he doesn't use summation notation, but it's kind of nice as well that the notation isn't as compact because it is easier to read at a glance. I always end up writing out and expanding whatever Lax has so compactly and elegantly written, simply because he leaves lots of proofs to the reader - which is great if the reader actually knows how to prove things. Not only does he assume that the reader is familiar with much mathematical notation (which he often does not bother to define), but also basics of proof-writing. Again, it's great if you're at that level, but it is a huge struggle for me even to comprehend what he's saying.
Yet in comparing Axler's book to the far more advanced style of Lax, I can see that this maturity is not only fluency in reading and interpreting the language of mathematics, including understanding the contextual use of symbols and typography that would otherwise be ambiguous, but the ability to fill in the blanks and appreciate not only whether a proof is, well, proven (still where I'm at!) but also whether it's beautiful as well. Axler chooses to write his proofs in a way that can be unwieldy but for me is often easier to understand. Lax's tend to be short and slick, but do not necessarily yield insight into the meaning of a theorem.
This "maturity" also seems to be about one's ability to make the leap to abstraction, having the ability to reinterpret the meaning of symbols dynamically (vertical bar meaning absolute value, norm, etc.) and also generalize their meaning in more powerful ways. I think it can include having a visual or intuitive understanding of concepts and - even harder - being able to express those in mathematical language, and choosing a formalism and notation that is clear and "natural" for one's purposes.
In other words, I don't get stuck on the meaning of x. I am okay with it being a number, a vector, a matrix, or other strange object as long as all of those follow the rules of typographical manipulation that I am allowed to practice on that object. I can erase old definitions and use new definitions for the same symbols (the vertical bar) without constantly calling the old definition. I don't panic when I see unfamiliar notation because I have confidence that reading the definition will clear things up.
I guess all of this is still pretty basic stuff, but it's about as far as I can imagine as a beginning mathematics student! I am definitely at the stage of learning a new language and all the grammar and cultural conventions that come with it. I constantly have to think about what I'm doing, try to translate my thoughts into the appropriate notation, and hope that I'm saying what I think I'm saying. I aspire to the kind of fluency in which I speak the language of mathematics and can focus more on what I want to say and express than the mechanics of doing so. I think another handy thing would be able to have a better vocabulary to describe mathematical ideas and notation. Because I'm such a visual thinker, I often have diagrams, pictures, or even movies in my head, but struggle to communicate verbally what I'm imagining. (Fortunately, I can draw.)
* * *
But as to mathematical maturity? My professor (in the last post) had a gut-level emotional reaction when he saw my (badly-written) equation because - I presume - he is fluent enough in mathematics that he just knew it was formulated incorrectly, even without being able to articulate why. It just looked wrong.
I'm still worlds away from that. The only situation in which I have that kind of fluency is in English. As a native speaker, my grammar may not always be correct, but I usually know when something is wrong - without any recourse to the appropriate grammar rule. I know because I've read and heard enough that the the familiar pattern of sentence structure is deeply ingrained, so it stands out when something does not fit that pattern of words and phrases. Perhaps someday I'll be fluent enough in the logic of mathematics that I will be able to use pattern recognition rather than the laborious conscious application of new ideas to read and understand theorems.
To sum it all up, Larry Deneberg says it most eloquently. For him, mathematical maturity is
... fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas.Something to aspire to, anyway.
---------------
* p. 79
(I rate my progress maturing mathematically by the length and intensity of this fear. It has declined from a solid minute of panic in the early days to a momentary blip because I know that if I dive in and read all the definitions, I will probably be able to understand what the statement means. Lesson 1 in mathematical maturity: Don't panic when seeing unfamiliar notation. After all, as a native English speaker, I don't panic when I see a word I don't know; I just look it up in a dictionary. But statements written in Russian seem pretty intimidating to me, as do many statements in mathematical language.)
***The inner product is also known as the scalar product or dot product.
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