Lobster-nomics

lobster pricesOne of the benefits of travelling around New England is enjoying lobster in its many forms:  steamed, on a roll, in a bisque, biting your toe.  But while shopping at a fishmonger one evening, I was somewhat baffled to see a sign like the one at the right.  It seems that the price per pound for lobster varies depending on the lobster’s size!

Why is this?  Why should a larger lobster cost more per pound than a smaller lobster?  Large or small, apples are still $1.69 per pound; the same goes for onions, chicken, and most other things.

What’s different about lobsters?  My assumption is that, in a lobster, the ratio of meat to non-meat (shell, antennae, veins, etc) is constant, that is, the same for any size lobster.  Thus, since you are paying for meat, you should then pay the same price per pound for any lobster.  But maybe that’s not the case.  Maybe in a small, one pound lobster, there is, say, 8 oz of meat and 8 oz of shell, but in a lobster twice the size, there is 20 oz of meat and 12 oz of shell.

lobsters

I guess I assumed that the geometry of the lobster is essentially the same regardless of the size; in other words, that all lobsters are geometrically similar.  Thus, the price-per-pound should be lobster-independent.  But maybe I’m wrong.  Is it really the case that, as lobster size increases, the amount of meat and the amount of non-meat increase at different rates?

Anyone have any other theories?

Passing Time and Tolls

coinsI enjoy traveling and I enjoy driving, but I don’t enjoy paying highway tolls.  One way I try to diffuse that annoyance is by hoarding coins when I travel, with the intent of paying every toll on the way back entirely in loose change.

I highly recommend the activity–it’s a fun little counting game, it keeps the coins in circulation, and there is a sense of satisfaction that accompanies using 18 assorted coins in a transaction.

But be warned–if you are travelling along the Hudson River in New York state, on I-87, and you decide to get off at Saugerties, one of the toll-booth operators there does not like pennies.  I mean, really, does not like pennies.  Especially more than 40 of them.  Trust me.

Socks and the Axiom of Choice

socksEvery time I buy socks I think of the Axiom of Choice.

About a century ago, mathematicians were arguing about exactly which basic axioms, or assumptions, were needed in order to justify all of our mathematics.  Because of the personalities involved and the nature of mathematical discourse at the time, Set Theory was the starting point, and one of the axioms under consideration was the Axiom of Choice.

Deciding on axioms is tough business:  an axiom has to be powerful enough to do something but obvious enough for people to accept it as true without evidence.  But deciding on axioms has to be done:  before we can prove anything, we need to assume something is true.

The Axiom of Choice essentially says that if you have an infinite number of sets, you can form a new set by choosing an arbitrary element from each of those sets.  It seems sensible enough, but fierce mathematical debate raged for years about whether this was obvious enough to be true.  Some mathematicians still don’t accept this principle.

So why would someone object to this sensible-enough idea?  That’s where shoes and socks come in.

Suppose you had infinitely many pairs of shoes.  There’s a straightforward way to define a new set that contains one shoe from each pair:  choose every left shoe.  This explicit rule make its clear how to construct this new set, and so forming this new infinite set seem reasonable.

But imagine you had infinitely many pairs of socks.  Since the socks are identical, you can’t give a specific rule that says “for each of the pair of socks, give me that one”.  You need to believe in the Axiom of Choice in order to believe that such a set, one containing one sock from each pair of socks, can really be formed without giving an explicit rule.

As it turns out, deciding to believe in this set of socks has substantial consequences for what you can prove in mathematics.  So there’s something to think about the next time you are sock shopping!

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