Wednesday, October 01, 2008

Mathematics and games, again

OK, I've had a chance to get back to wrestling with Sec. 108 of the Big Typescript. So here's a draft handout for the first seminar of term -- mostly for third year undergrads and beginning post-grads (so this is neither very detailed nor very sophisticated; but I hope it is at least comprehensible and will provoke some discussion). Comments very welcome though!


Christopher F. S. Maligec said...

An abacus is designed for practical calculation and not for logical propositions about arithmetic. The difference is that the latter allows for negation, with the corresponding symbol, and the former doesn't. To adapt, one only needs to allow for a negating operation on an abacus. For example, add a new "negation row" such that a "negative" bead is moved from right to left if an incorrect move has been made in the rows for numbers. Then, 3+4=8 would have one negative bead on the negation row and the end result is a true proposition. In the case of multiple negations, every two negative beads cancel out. An example, "3+4=7 with one negative bead" is false, but the movement of the negative bead was incorrect, so a second negative bead is moved to note this and the two cancel out, leaving the correct proposition.

Similarly, if negation and its symbol were elininated from the logic of arithmetic, you would have the same problem as with the original abacus.

In short, the use of an abacus or graphic symbols doesn't matter; what matters is the abstract system being represented (or "embodied"). If the abstract system is geared toward practical, numerical calculation, then it wouldn't have negation, and if it were geared toward a logical number theory, it would.

Peter Smith said...

I agree with the drift of these remarks. We could envisage a use of an abacus-frame where there was an analogue of negations; and of course we can envisage negation-free fragment of arithmetic. The question is what makes the difference between practices with and without negation. And just saying it is "application" in deriving new non-arithmetical propositions from old ones. For both types of practice could be used for that.

Christopher F. S. Maligec said...

Perhaps the difference could be ascribed to objective.
A non-negative fragment better serves the commercial and day-to-day arithmetical purposes that an abacus was designed for, as a fuller system would overgenerate propositions of no relevance. The same fragment, however, is incomplete with respect to a logical theory of arithmetic and fails to achieve its objectives.

I have a question about the definition of a game. Do all games require competition with criteria for a winner(s) (e.g. the first to obtain the winning state, the player with the largest payoff or the least loss)? Are single -player games really games? (Is solitaire a game or really a problem to be solved?)

Or are games those activities the main purpose of which is to achieve entertainment? In that case "the first to prove a new theorem or to invent a new machine" would be games even though they produce mathematics or a practical new machine as a by-product.

P.S. Would this be correct:
Propositions take on truth values whereas moves take on "good" values? Where a good move is one that increases the chances of winning and a bad one is one that reduces or does nothing to the chances of winning.

a cartesian said...

Surely if we're talking Wittgenstein, there can't be anything such as being competitve or having a winner that's true of all games, and instead there has to be only a "family resemblance" among games.

(Perhaps some triviality is true of all games - perhaps (guessing) they're all activities of some sort, or are called a game at least in some language by somebody somewhere, but nothing more "interesting".)