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The high school teacher in the link is a B.S. in math education. They're usually reflexive Platonists, believing that math is out there, and we merely discover

The high school teacher in the link is a B.S. in math education. They're usually reflexive Platonists, believing that math is out there, and we merely discover it. This is a result of the teaching of undergrad math as essentially a series of completed works, with little history attached to it. This teacher probably hasn't thought critically about why, say, 1/x^2 = x^(-2). By contrast, a mathematician has a Ph.D. in math, and has had to do original research and look at some history of topics. Thus, the mathematician knows that math is not a finished product, but is under constant refinement. Now, let's talk about negative exponents. Students are taught that 1/x^2 = x^(-2). High school teachers often don't understand why. It is so ingrained that even asking "Why?" seems almost grammatically incorrect. The reason why is that we know the following things about exponents: 1) x^n = x* x* ...* x for n a positive integer 2) As a consequence of 1, x^m* x^n = x^(m+n) for m,n positive integers Now, the question is not "what is x^n if n is negative?" (which is what a high school teacher might ask). Rather, the question is "Can we define (!) x^n for negative n in a way consistent with the item two above?" (mathematician's framing). And, of course, we can. If x^n = 1/x^(-n) for n negative, then item two works. So, a high school teacher most likely thinks that the negative exponent rule is simply a rule, handed down from the Gods of math. A mathematician recognizes that it is a convention, and such a smooth convention that there is simply no better choice. Now, about 0^0: The HS teacher asks "What is 0^0?" and is therefore under the impression that 0^0 is undefined because according to certain reasonings it could be 0 or it could be 1. Textbooks (not written by mathematicians) wouldn't correct this. The TI-86 gives a domain error when 0^0 is input. The mathematician asks "What value of 0^0 makes my preferred formulae continue working?" and thereby defines 0^0=1.

jpadkins 15y ago
wait.. you mean math is simply made up? It's not the language of the universe? well that's just depressing.
- sliverstorm 15y ago
  Although my mathematician friends would probably yell at me if they heard this, as far as I see things it's hardly any different from physics. Just a model we create to describe observed phenomenon.
  - xyzzyz 15y ago
    Oh, math only works this way in intuitive fields, like elementary calculus, elementary probability and staticstis, graph theory or Euclidean geometry. However, there are fields in math that are different -- for instance, there are topological spaces that exhibit phenomenons unseen anywhere else and that are very hard to grasp intuitively (I had very hard time trying to imagine what Cech-Stone compactification construction based on ultrafilters look like, and eventually I gave up after I understood that this space is only supposed to satisfy an universal property, and not have any intuitive shape). Deep results in algebraic methods in topology say interesting things about the models, while being completely indescribable outside of them, apart from some trivial examples (for instance, what _exactly_ are cohomology classes?). The bottom line is, if the only things you've seen are geometry and calculus, with intuitive concepts like speed of change, area, length, then yeah, it's only making our intuitions more formal. Otherwise, it's something completely different.
    - sliverstorm 15y ago
      Again, I would argue this is no different than physics. The strength and maturity of a model is not only based on its ability to describe what we can verify, but predict what we have never (and may never) seen, and sometimes what we have never even conceived of- and I think both physics and mathematics are doing that.
      - xyzzyz 15y ago
        My point is, many concepts in math do not exist outside of the models describing them, while if we throw away physical models, the reality does not go away.
    - Almaviva 15y ago
      I'm exactly the intuitive thinker you describe, with only an undergrad degree in math. I have not a clue what a Cech-Stone compactification construction based on ultrafilters is. But... I think that whatever it is, you decide on some fairly simple properties you want to satisfy, and then go off discovering what they lead to and what the consequences are. Sometimes (mostly all the time?) you get something trivial or that reduces to being isomorphic to something else, but sometimes you get out a lot more than you put in, in surprising ways. I call that a lot more like "discovery" than "invention" though both are strained as analogies.
- cperciva 15y ago
  God created the integers. All else is the work of man.
noahl 15y ago
> They're usually reflexive Platonists, believing that math is out there, and we merely discover it. This is a result of the teaching of undergrad math as essentially a series of completed works, with little history attached to it. It's not entirely clear what you mean by this, but in the most obvious interpretation this idea is correct. Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet. The other interpretation, which you probably meant, was the question of mathematical style and interest: which axioms do we study, and why do we care about them, and why are some theorems important and others not important? Certainly the definition that 1/x^2 = x^(-2) was made for consistency, which is an aspect of style. You could make some other definition and do formal reasoning just as well (although you'd have to change some other things - maybe the definition of exponentiation, maybe your interpretation of symbols on paper). But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students.
- yaks_hairbrush 15y ago
  > Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet. From one viewpoint, math deals with pure logic. It's something of a poor viewpoint from my perspective: Newton couldn't back up his calculus with logic, Euler and Riemann made numerous unfounded assumptions when looking at the zeta function, Heaviside built a telegraph across the Atlantic despite lacking a proper logical foundation. My personal viewpoint is that mathematics is a series of shortcuts for understanding and manipulating a wide variety of phenomena, and mathematical research is the development of further shortcuts. Often logic comes in, but it's usually after you get the result. Nothing wrong with logic (I sure do like computers, for example!), it's just not the panacea for anything mathematical. > But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students. You're absolutely right about this, and I was specifically thinking of two very bad high school teachers when I wrote my post. The fact is that I know very few HS math teachers, and so my opinion is quite clouded by these two.
Almaviva 15y ago
Math is out there. I don't understand how anyone can say that the Mandelbrot set was created or is formed from arbitrary axioms. It was discovered full stop.
- jberryman 15y ago
  Your parent never says anything about arbitrary axioms. To your main point: are irrational numbers, say, "out there"? If so, where? Until a few centuries ago it was mathematical standard practice to fudge 1/2 as 25/49 when taking it's square root. But then mathematicians invented (some would argue) the notion of an irrational, because it was, well... useful. There are real metaphysical questions here; there have been since the greeks started reaching the limits of a purely geometric (read physical) understanding of math.
  - Almaviva 15y ago
    I don't think the word "invented" is much better an analogy for real numbers than "position" is for "where" the Mandelbrot Set is. This goes beyond what is strictly math, but I don't think it's reasonable to say that properties of the real numbers (say, roots, pi, e, and so forth) are invented. In some sense they seem like the simplest thing that fits a few properties (and not that many). Similarly with Euclidean Geometry, and the natural numbers (with primes and their structure, etc.) I really actually think something akin to Occam's Razor applies in math! You can of course tweak your starting points and get really interesting things too, say non-standard analysis or non-Euclidean geometries. And you can derive (I would say "discover") some rich and surprising properties and patterns in them. But I don't think any of these things are accidental. The real numbers (and pi, e, etc) would surely be "invented" in the same (modulo shifts in convention) ways by other advanced civilizations, I think. Could you imagine this not being so? This is also extra-mathematical, and I admit this may be hooey, but I believe (as a non-mathematician) that math gives hints at large patterns and tells you when things fit well or are funny. Like, I can argue whether Pi or 2*Pi is the more "natural" constant, and it's not a discussion completely devoid of content! Also, I think math tells me something is funny about 0^0 (because of the limit 0^x) but that 1 is the more natural fit. I realize this puts me quite firmly into the category of people being belittled here!
dataduck 15y ago
Not entirely fair. While there is a lot of choice about how to define stuff around the edges of mathematics, after the axiomatic choices have been made, there's quite a lot to discover in their structure. Particularly if the relevant axioms relate to something outside of mathematics (the examples are too many to even scratch but start out thinking of the definitions of natural and rational numbers) then you really are discovering things in the same way as a physicist - in fact, this is how many physicists go about discovering things, for better or worse. In most situations it makes sense to define exponentiation as repeated multiplication, and a^0 as the absence of multiplication by a, hence a^0 = 1 as the multiplicative identity. I wouldn't introduce the idea of anything else to a student unless they specifically asked me about one of the problems which can arise in choosing 0^0 = 1.

jpadkins 15y ago

wait.. you mean math is simply made up? It's not the language of the universe? well that's just depressing.

sliverstorm 15y ago
Although my mathematician friends would probably yell at me if they heard this, as far as I see things it's hardly any different from physics. Just a model we create to describe observed phenomenon.
- xyzzyz 15y ago
  Oh, math only works this way in intuitive fields, like elementary calculus, elementary probability and staticstis, graph theory or Euclidean geometry. However, there are fields in math that are different -- for instance, there are topological spaces that exhibit phenomenons unseen anywhere else and that are very hard to grasp intuitively (I had very hard time trying to imagine what Cech-Stone compactification construction based on ultrafilters look like, and eventually I gave up after I understood that this space is only supposed to satisfy an universal property, and not have any intuitive shape). Deep results in algebraic methods in topology say interesting things about the models, while being completely indescribable outside of them, apart from some trivial examples (for instance, what _exactly_ are cohomology classes?). The bottom line is, if the only things you've seen are geometry and calculus, with intuitive concepts like speed of change, area, length, then yeah, it's only making our intuitions more formal. Otherwise, it's something completely different.
  - sliverstorm 15y ago
    Again, I would argue this is no different than physics. The strength and maturity of a model is not only based on its ability to describe what we can verify, but predict what we have never (and may never) seen, and sometimes what we have never even conceived of- and I think both physics and mathematics are doing that.
    - xyzzyz 15y ago
      My point is, many concepts in math do not exist outside of the models describing them, while if we throw away physical models, the reality does not go away.
  - Almaviva 15y ago
    I'm exactly the intuitive thinker you describe, with only an undergrad degree in math. I have not a clue what a Cech-Stone compactification construction based on ultrafilters is. But... I think that whatever it is, you decide on some fairly simple properties you want to satisfy, and then go off discovering what they lead to and what the consequences are. Sometimes (mostly all the time?) you get something trivial or that reduces to being isomorphic to something else, but sometimes you get out a lot more than you put in, in surprising ways. I call that a lot more like "discovery" than "invention" though both are strained as analogies.
cperciva 15y ago
God created the integers. All else is the work of man.

noahl 15y ago

> They're usually reflexive Platonists, believing that math is out there, and we merely discover it. This is a result of the teaching of undergrad math as essentially a series of completed works, with little history attached to it. It's not entirely clear what you mean by this, but in the most obvious interpretation this idea is correct. Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet. The other interpretation, which you probably meant, was the question of mathematical style and interest: which axioms do we study, and why do we care about them, and why are some theorems important and others not important? Certainly the definition that 1/x^2 = x^(-2) was made for consistency, which is an aspect of style. You could make some other definition and do formal reasoning just as well (although you'd have to change some other things - maybe the definition of exponentiation, maybe your interpretation of symbols on paper). But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students.

yaks_hairbrush 15y ago
> Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet. From one viewpoint, math deals with pure logic. It's something of a poor viewpoint from my perspective: Newton couldn't back up his calculus with logic, Euler and Riemann made numerous unfounded assumptions when looking at the zeta function, Heaviside built a telegraph across the Atlantic despite lacking a proper logical foundation. My personal viewpoint is that mathematics is a series of shortcuts for understanding and manipulating a wide variety of phenomena, and mathematical research is the development of further shortcuts. Often logic comes in, but it's usually after you get the result. Nothing wrong with logic (I sure do like computers, for example!), it's just not the panacea for anything mathematical. > But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students. You're absolutely right about this, and I was specifically thinking of two very bad high school teachers when I wrote my post. The fact is that I know very few HS math teachers, and so my opinion is quite clouded by these two.

Almaviva 15y ago

Math is out there. I don't understand how anyone can say that the Mandelbrot set was created or is formed from arbitrary axioms. It was discovered full stop.

jberryman 15y ago
Your parent never says anything about arbitrary axioms. To your main point: are irrational numbers, say, "out there"? If so, where? Until a few centuries ago it was mathematical standard practice to fudge 1/2 as 25/49 when taking it's square root. But then mathematicians invented (some would argue) the notion of an irrational, because it was, well... useful. There are real metaphysical questions here; there have been since the greeks started reaching the limits of a purely geometric (read physical) understanding of math.
- Almaviva 15y ago
  I don't think the word "invented" is much better an analogy for real numbers than "position" is for "where" the Mandelbrot Set is. This goes beyond what is strictly math, but I don't think it's reasonable to say that properties of the real numbers (say, roots, pi, e, and so forth) are invented. In some sense they seem like the simplest thing that fits a few properties (and not that many). Similarly with Euclidean Geometry, and the natural numbers (with primes and their structure, etc.) I really actually think something akin to Occam's Razor applies in math! You can of course tweak your starting points and get really interesting things too, say non-standard analysis or non-Euclidean geometries. And you can derive (I would say "discover") some rich and surprising properties and patterns in them. But I don't think any of these things are accidental. The real numbers (and pi, e, etc) would surely be "invented" in the same (modulo shifts in convention) ways by other advanced civilizations, I think. Could you imagine this not being so? This is also extra-mathematical, and I admit this may be hooey, but I believe (as a non-mathematician) that math gives hints at large patterns and tells you when things fit well or are funny. Like, I can argue whether Pi or 2*Pi is the more "natural" constant, and it's not a discussion completely devoid of content! Also, I think math tells me something is funny about 0^0 (because of the limit 0^x) but that 1 is the more natural fit. I realize this puts me quite firmly into the category of people being belittled here!

dataduck 15y ago

Not entirely fair. While there is a lot of choice about how to define stuff around the edges of mathematics, after the axiomatic choices have been made, there's quite a lot to discover in their structure. Particularly if the relevant axioms relate to something outside of mathematics (the examples are too many to even scratch but start out thinking of the definitions of natural and rational numbers) then you really are discovering things in the same way as a physicist - in fact, this is how many physicists go about discovering things, for better or worse. In most situations it makes sense to define exponentiation as repeated multiplication, and a^0 as the absence of multiplication by a, hence a^0 = 1 as the multiplicative identity. I wouldn't introduce the idea of anything else to a student unless they specifically asked me about one of the problems which can arise in choosing 0^0 = 1.