I join the blagosphere

For the discussion of math. Duh.

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I join the blagosphere

Postby SunAvatar » Sun Oct 03, 2010 4:39 pm UTC

For a long time I've had a sort of vague plan to start some sort of expository mathy blog for laypeople. Plan A was to write about some specific topic like calculus, but I found that I couldn't plan that far ahead. So Plan B is now in action!

Irrational Inquiries is an expository math blog for lay people. I'm endeavoring to assume as little background knowledge as I can get away with. (The ideal reader will be comfortable with the arithmetic of the rational numbers, but this may not turn out to be strictly required.) In the next however-many posts, I hope to write about
  • axiomatic treatments of the natural numbers, integers, rational numbers, and real numbers;
  • constructions of the same within set theory;
  • counting infinite sets; and
  • the notion of continuity.
I emphatically don't have a time-frame planned, lest I tempt Hofstadter's Law. I will go as slowly as I must.

Hopefully the posts (well, post currently) will be of interest to some folks here. There are a lot of people here for which everything will be very elementary, but I'm hoping those people might still be willing to offer suggestions/criticisms/mockery. I've never really tried to maintain a blog before, and so I'm pretty much making it up as I go along.
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RogerMurdock
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Re: I join the blagosphere

Postby RogerMurdock » Sun Oct 03, 2010 5:37 pm UTC

Sounds pretty awesome? I read your first entry and it was extremely interesting. I have a background in Calculus and knew some general ideas about sets but the entry was very informative and easy to understand. I probably actually *gasp* learned something.

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Re: I join the blagosphere

Postby imatrendytotebag » Sun Oct 03, 2010 5:49 pm UTC

If all of your entries are as lucid and enjoyable to read as the one you've posted, this blog is going to be awesome.
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Re: I join the blagosphere

Postby willancs » Sun Oct 03, 2010 7:07 pm UTC

Excellent - really enjoyed reading that and while I already had an idea of the construction of the counting numbers from sets, your post sets it out clearly and vividly - great job!

One question - have you got an RSS feed anywhere on there? Would love to catch your (potential!) future posts, as mathematics is sadly under-represented in my blog reading.

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Re: I join the blagosphere

Postby SunAvatar » Sun Oct 03, 2010 8:48 pm UTC

RSS feed, good idea! I've added the relevant widget. (Like I said, no experience in this blagging stuff.)
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Re: I join the blagosphere

Postby Talith » Sun Oct 03, 2010 11:24 pm UTC

Really enjoyed the first post. I've subscribed to the rss and I think I may introduce some friends that aren't mathematically inclined as I've tried to explain bijections, set theory and cardinalities before but I think I do a pretty shoddy job of it; you certainly put into words what I'd really like to communicate to them.

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Re: I join the blagosphere

Postby mdyrud » Mon Oct 04, 2010 2:38 am UTC

That was fantastic. I am definitely looking forward to more posts in the future. It was so good, it made my Facebook wall, which is a pretty big deal. Good job!

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Re: I join the blagosphere

Postby SunAvatar » Mon Oct 04, 2010 3:25 am UTC

Everyone seems pretty happy with it so far! Or else the people who aren't are very polite.

It's pretty imperative that I move away from the computer now and try to get some real work done, but I just wanted to point out another post is going public at midnight. Since I have a lot of ideas and am still pretty excited right now, I'm trying to get ahead of schedule.

"Schedule" is tentatively planned to be three posts a week, if I can keep up with that without sacrificing the quality too much. (Quantity, on the other hand---that was a pretty long post, and I might have to break things up a bit more in the future.)
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Re: I join the blagosphere

Postby Amaris » Mon Oct 04, 2010 4:49 am UTC

Very lucid and engrossing. I did notice a couple of errors ("...if a ≤ b and b ≤ c, then a ≤ b."), but other than that, I'd give it an A++.

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Re: I join the blagosphere

Postby SunAvatar » Mon Oct 04, 2010 5:07 am UTC

Oops, fixed. I'm re-proofreading now; what other errors did you notice?

(And yeah, apparently I just can't stay away.)
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Re: I join the blagosphere

Postby Divinas » Mon Oct 04, 2010 8:50 am UTC

It is insanely awesome. Keep up the good work, I sincerely hope you won't give up.

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Re: I join the blagosphere

Postby Qaanol » Mon Oct 04, 2010 8:59 am UTC

I very much like this. Especially this:

Mathematicians have mostly solved these problems by adding more rules than the ones above, putting technical provisos and restrictions on exactly how you can define a set. We will solve these problems by ignoring them, and staying away from the situations where they would come up.
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Re: I join the blagosphere

Postby Amaris » Mon Oct 04, 2010 9:43 pm UTC

SunAvatar wrote:Oops, fixed. I'm re-proofreading now; what other errors did you notice?


Huh. I can't seem to find them anymore. Probably just something I imagined.

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Re: I join the blagosphere

Postby Syrin » Mon Oct 04, 2010 10:18 pm UTC

You put the odd numbers before the even numbers in one of your examples, but said the opposite.

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Re: I join the blagosphere

Postby SunAvatar » Tue Oct 05, 2010 12:11 am UTC

What? Did not. Just look at the post if you don't believe me!


(just kidding, fixed)
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Re: I join the blagosphere

Postby SunAvatar » Thu Oct 21, 2010 3:24 am UTC

Hey I hate to double post, but I asked a question of my audience and I figure I should ask here too.

What should I write about next:
  • constructing the integers/rationals/etc.,
  • infinite cardinal numbers, or
  • something else?
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nehpest
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Re: I join the blagosphere

Postby nehpest » Thu Oct 21, 2010 5:10 am UTC

Oooh, let's hear about integers/rationals/reals next. Please?
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Re: I join the blagosphere

Postby Syrin » Thu Oct 21, 2010 6:01 am UTC

Constructing the integers/rationals/reals is an interesting topic. "Bam dedekind cuts, surprise you now have the reals"
It's pretty cool.

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Re: I join the blagosphere

Postby skeptical scientist » Thu Oct 21, 2010 6:05 am UTC

I think you should define integers as equivalence classes of ordered pairs of naturals, and rationals as equivalence classes of ordered pairs of integers, and then tell us what set the number 1/2 is by listing its elements.

I hear that people don't like fractions, and I think really explaining what they are in this way is sure to fix that.
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Re: I join the blagosphere

Postby tckthomas » Thu Oct 21, 2010 6:40 am UTC

what happened to division and subtraction?

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Re: I join the blagosphere

Postby SunAvatar » Thu Oct 21, 2010 6:43 am UTC

skeptical scientist, I was actually planning something like that---except that I was going to first talk about what the integers "should be," axiomatically (the closure of the natural numbers under subtraction). This may at some point result in a digression into rigs and rings and stuff.

Really, the construction of the integers, rationals, algebraic numbers, and real/complex numbers are all kind of the same trick: you name the closure properties you want your extension to have (subtraction, division, polynomial roots, or sequence limits), then define the objects of the extension in the dopiest way that could possibly work (as equivalence classes of formal subtraction-pairs, formal division-pairs, polynomials, or sequences).

I've never really understood why people don't like fractions, even though I've seen it to be the case (even with college students who really ought to have gotten over that one hurdle by now). But then my parents were explaining fractions to me while cutting my sandwich when I was still too young to use a knife myself, so that probably has more to do with my understanding than any real cleverness on my part.

tckthomas, subtraction is basically "solving an additive equation." Perhaps I should write more about that.
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Re: I join the blagosphere

Postby tckthomas » Thu Oct 21, 2010 7:33 am UTC

yay! subtraction! :D

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Re: I join the blagosphere

Postby skeptical scientist » Thu Oct 21, 2010 8:01 am UTC

SunAvatar wrote:skeptical scientist, I was actually planning something like that---except that I was going to first talk about what the integers "should be," axiomatically (the closure of the natural numbers under subtraction). This may at some point result in a digression into rigs and rings and stuff.

That was a joke, to be honest. I think that is the worst possible way to understand what negative numbers/fractions are.
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Re: I join the blagosphere

Postby SunAvatar » Thu Oct 21, 2010 8:22 am UTC

Wow, really? I thought the worst possible way to understand what negative numbers were was "negative times negative is positive or you fail the class, bucko."

What do you think integers are? I guess I agree that they're not "really" equivalence classes of pairs of natural numbers. But then natural numbers aren't "really" finite von Neumann ordinals, either---it's just a neat construction that (hopefully) is somewhat based on what we want them for.

Hm... is there a nice setting in which the integers can be considered as representatives of isomorphism classes of some objects, the way that natural numbers are representatives of bijection classes of sets? That would be spiffy.
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Re: I join the blagosphere

Postby Syrin » Thu Oct 21, 2010 11:16 am UTC

I actually like how naturals->integers->rationals progresses from monoid->group, give it multiplication and then it's a ring->field. Then every step of the way seems like a natural extension from the previous - so, yeah, closure under subtraction (additive inverses) and division (multiplicative inverses). Looking at them as equivalence classes is a useful but not (in my opinion) very satisfying method. There doesn't really seem to be a reason for it.


Actually something I would suggest because I only just thought of it:

Do the infinite ordinals! I'm sure that there must be one or two new readers unfamiliar with math who look at the von Neumann construction of the naturals and sit down and ponder about the set of the natural numbers itself. I know I did when I first encountered it.

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Re: I join the blagosphere

Postby jestingrabbit » Thu Oct 21, 2010 12:14 pm UTC

SunAvatar wrote:What do you think integers are?


They're the naturals, with (0 and) the negatives of the naturals appended. Defining addition and subtraction from there isn't too bad, and then explaining that subtracting is adding the negative follows pretty naturally from there.
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Re: I join the blagosphere

Postby skeptical scientist » Fri Oct 22, 2010 2:29 am UTC

I guess I usually think of the naturals, integers, rationals, and reals as being defined by their properties. So the naturals are an inductive ordered semiring (inductive meaning they satisfy full 2nd-order induction), the integers are the ring generated by the naturals, the rationals are the field generated by the integers, and the reals are the ordered field containing the rationals and satisfying the least upper bound property. Formal constructions are useful to show that a structure exists which satisfies the given properties, but are then best forgotten, since it really doesn't make a whole lot of sense to identify the square root of two with an infinite set of rational numbers, or with an equivalence class of Cauchy sequences. I also tend to think of other fields isomorphic to the reals as still being (in a sense) "the reals", and similarly for the other structures.
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Re: I join the blagosphere

Postby Black » Fri Oct 22, 2010 1:47 pm UTC

Structuralism is awesome. Constructing models is good to show that your structure makes sense. We assume that set theory is consistent. So, if you construct the reals in set theory by any one of the many constructions then you know it makes sense: the theory is consistent relative to your mathematical framework.

I like to think of things w.r.t. universal properties. The natural numbers are the initial algebra of the signature Z : 1 -> X and S : X -> X. This initiality gives them induction principles.

The integers are then the Groethendieck group of the naturals considered as a monoid.

The rationals are the field of fractions of the integers.

The reals are the completion of the rationals considered as a metric space.

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Re: I join the blagosphere

Postby SunAvatar » Sat Oct 23, 2010 5:19 am UTC

That is a good structural characterization of the natural numbers; another nice one is "the initial semiring" (or possibly "the free semiring on no generators"). I don't really like defining the natural numbers to be "something you can do induction with," since induction as a concept is rather newer than the natural numbers.

This also works with the integers, a.k.a. the initial ring. Unfortunately this characterization breaks down a bit with the rational numbers.... ah, well, it's nice when it works.

I guess I agree that the equivalence-class thing is more of an "implementation" than it is an actual fact about what integers/rationals/reals are. But the implementation itself is pretty cool! The fact that you can use that one damn trick over and over again---that you can invent answers to questions, just by defining equivalence classes of the questions you want to answer---seems pretty deep to me, and may possibly even be *gasp* useful to know.
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