First, I’ll mention that I’ve updated my blogroll – there’s been a real burst in math blogs over the summer, at least in part instigated by my friends at the Secret Blogging Seminar, but also by the spurt of Fields Medalists with blogs. (Are we up to 10% of the total number now?) I’ve also added a few philosophy blogs that I’ve been reading for a while, and a couple that I should have been reading, but of course I’m sure I’m missing others.

Anyway, there’s new math job search gossip stuff going on on the web – I think the discussion on that post is interesting and relevant across disciplines for people trying to figure out whether this is generally a good thing or not.

Tim Gowers discusses the way logarithms and other abstract things should be taught. He advocates a way that’s a bit more formalist than some others suggest, but it sounds reasonable to me. There’s also interesting discussion of formalism there in the comments, though some of it sounds more like structuralism to me. See for example Terence Tao’s comment, “I guess there is a fundamental transition in mathematical learning when one realises that what mathematical objects *are* (and how they are constructed) may be less important than what mathematical objects *do* (e.g. what properties they obey).”

Also, a discussion about the Axiom of Choice at The Everything Seminar (I may add that one to my links later too), focusing on a puzzle I first heard from my friend Lukas Biewald. There’s interesting discussion in the comments that reveals implicit ideas about platonism and formalism among mathematicians. I think the anti-platonist majority there should be a bit more careful though, because similar issues apply in arithmetic, thanks to Gödel’s results. I think we should be much more hesitant to say that the natural numbers are just something we make up than they are with the universe of ZFC (or a topos, or whatever), as I mentioned before.

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Enigman(06:39:46) :Hi, thanks for that last link, which I’d missed (and was indeed interesting:)

Todd Trimble(23:28:47) :“I think we should be much more hesitant to say that the natural numbers are just something we make up than they are with the universe of ZFC (or a topos, or whatever), as I mentioned before.”

As a member of the “anti-platonist majority” (I think there were only two or three of us posting, hee!), I actually agree with what I think you’re saying. There’s a universal property of the usual natural numbers, a higher-order property if you like, which characterizes it up to isomorphism. It simply says that N is initial among pointed sets equipped with an endomorphism. That’s how I always think of N.

It’s a nice uniform definition which works for any topos or “category of sets”, and logical morphisms between toposes preserve natural number objects. So you really could say, in the sense of parametric polymorphism, there is only one true N! In their nice book on categorical logic, Lambek and Scott take this *sort of thing* as a possible resolution between Platonism and other philosophical positions, and that seems pretty reasonable to me.

Todd Trimble(23:32:09) :Bah, I should have said sets equipped with a point and an endomorphism (the endomorphism need not preserve the distinguished point, obviously).

Kenny(20:55:43) :That’s good to hear that the natural number object is preserved under morphisms from one topos to another, but this fact about there being “one true N” doesn’t seem quite sufficient. After all, if there’s only “one true N”, a natural question would be whether or not it contains a solution to any given diophantine equation. But by Gödel’s work (and the applications of it Davis/Putnam/Robinson/Matiyasevich made to Hilbert’s 10th Problem), no recursive set of axioms (such as the axioms describing a topos) will settle all these questions. Thus, this “one true N” will have some representatives in which a given diophantine equation will have solutions, and some representatives in which it won’t.

Todd Trimble(22:41:27) :Kenny, that’s an interesting point you make. But even though there is a uniform definition-up-to-isomorphism of “true N” internal to a topos doesn’t mean that the external set of points hom(1, N) will be the same in all cases, so there may be a solution 1 –> N of a diophantine equation in one topos which doesn’t exist in another. I am thinking particularly of the fact that there is a filter-quotient construction Set//U wrt an ultrafilter U, so that for the internal N in this topos, the set hom(1, N) is precisely the corresponding ultrapower of N. So yes, there is some fine print involved.

Ryan Streitenberger(21:33:40) :I’m looking for a mathmetical probability solution to a drawing probability for a special situation of drawing for random numbers.. example I have 408 poeple as members of a club; typically we would have an individual number for each person on a list of membership. However, we cannot guarentee each number is physically present. Therefore, we implemented a new technique.

We draw a single digit at a time.

First digit, one number 0 thru 4 would be drawn.

Second digit 0 thru 9, with one contingency: if a 4 is drawn on the first digit, the second is automatically a 0, then the third digit is drawn for a 0 thur 8.

If not, 0 thru 3 is drawn, then all ten numbers is drawn from, then third digit is drawn.

Third digit, is drawn from.

This is the fair method in which I believe in.

The opposing method is that if a 4 is drawn on the first digit, then all ten numbers are drawn from. If a 1 thru 9 is drawn, then both the 4 and the second number is thrown out, (because you are looking for a number from 1 to 408, only)

When a 4 is drawn on the first digit, a zero, must be drawn on the second for the drawing to continue. If not all number are thrown out.

Then you begin with the first digit again.

My argument is that is my number is 406, then I have a 9 in 10 chance that my number 4-hundred is eliminated after the first draw by the second draw, therefore, if I’m in the 4-hundreds, I have just as much of a chance to be eliminated on the second draw as the first. Thus my number has less of a chance to come up after a recurring draw. I am taking into accountability of the actual draw as a reduction of odds upon that draw. We’re only going to draw a single number through 408, not searching for the particular draw occurance.

Can anyone explain this in a better way???

Kenny(10:33:15) :On the first way, let’s consider the probability that either person 008 or person 408 gets chosen. For person 018, there’s a 1/5 chance that 0 is drawn first, then a 1/10 chance that 1 is drawn second, and then a 1/10 chance that 8 is drawn third, so the overall chance of getting picked is 1/500. Now for person 408, there’s a 1/5 chance that 4 is drawn first, then a 1/1 chance that 0 is drawn second, then a 1/9 chance that 8 is drawn third, so there’s an overall 1/45 chance of getting picked. Thus, this first method gives different numbers different probabilities.

On the second method, everyone gets a 1/500 chance of getting picked on the first draw, and if the first draw doesn’t pick anyone (which has a 92/500 chance of happening), then we repeat the process, again with everyone having the same chance of getting picked. Thus, the second method gives everyone an equal chance.