There's this game I sometimes set up...

hang on...

here. https://thephilosophyforum.com/discussion/11547/bannos-game/p1

It'll be moved to the lounge as soon as the mods see it, and disappear.

But it seems to me to be a good metaphor for the development of both language and mathematics. More a simplified case than a metaphor.

Folk can add rules - and quickly they will construct paradoxes, but after a dozen or so posts the rules will branch so that some folk are following some rules but not others, sometimes giving justifications for the path they choose, sometimes not.

That's what maths is - sets of rules we have made up, with folk building on the ones they find interesting.

You can play a variety of chess with new rules that the queen can move only two squares in any direction and there are no pawns. But you can't do any kind of arithmetic by stipulating that 3 + 5 does not equal 8. That's because queens and pawns are our constructions and a queen is and does whatever we say and without us there is no such thing as a queen. But there were two atoms of hydrogen to one of oxygen in water before we came along. If there hadn't been we couldn't have evolved to learn to count them.

Interesting argument indeed. It is true that the queen only can move according how the rules are established but this is just a basic axiom to pursue equity of opportunities of win between both players. You and me (if we want to) can change the rules in a private play without competition.

Back to mathematics, when you said "5 + 3 equals 8", it reminds me an interesting video that probably you would like about how free of interpretation are our sum orders.

Ha ha! very good - this is the same topic in a bit more depth

The first video (I didn't watch the second video) is stupid nonsense and disinformation.

In this context, infinite summation is defined only for converging sequences. If the rules of definition are violated, then, of course, contradictions may be derived. There is no mystery or even problem about that.

The person at the blackboard says, "The problem with infinity is all sorts of weird things happen when you're dealing with infinity". First, that doesn't even mean anything. Second, instead of explaining that the fallacy is in using an undefined notion (infinite summation on a sequence that does not converge), the person at the blackboard doesn't even suggest how we may investigate further to see that there is not an actual conundrum.

The video is yet another example of Internet ignorance and disinformation. That person seems to be teaching a classroom. He should be told by the school administrators to clean up his act: If he wants to present mathematical challenges, then he should provide his students with the benefit of techniques and information for solving the challenges rather than obfuscate with "weird things happen".

The first video (I didn't watch the second video) is stupid nonsense and disinformation.

In this context, infinite summation is defined only for converging sequences. If the rules of definition are violated, then, of course, contradictions may be derived. There is no mystery or even problem about that.

The person at the blackboard says, "The problem with infinity is all sorts of weird things happen when you're dealing with infinity". First, that doesn't even mean anything. Second, instead of explaining that the fallacy is in using an undefined notion (infinite summation on a sequence that does not converge), the person at the blackboard doesn't even suggest how we may investigate further to see that there is not an actual conundrum.

The video is yet another example of Internet ignorance and disinformation. That person seems to be teaching a classroom. He should be told by the school administrators to clean up his act: If he wants to present mathematical challenges, then he should provide his students with the benefit of techniques and information for solving the challenges rather than obfuscate with "weird things happen". — TonesInDeepFreeze

:up: :up: :up: :up: :up:

The first video (I didn't watch the second video) is stupid nonsense and disinformation. — TonesInDeepFreeze

Sorry. Mi fault for sharing it, I thought it could be interesting.

You can play a variety of chess with new rules that the queen can move only two squares in any direction and there are no pawns. But you can't do any kind of arithmetic by stipulating that 3 + 5 does not equal 8. That's because queens and pawns are our constructions and a queen is and does whatever we say and without us there is no such thing as a queen. But there were two atoms of hydrogen to one of oxygen in water before we came along. If there hadn't been we couldn't have evolved to learn to count them. — Cuthbert

Very nice comeback - thanks! ( I assume it was directed to me? I'll read it that way.)

You can't play chess unless you stipulate a distinction between pawns and queens. That's more on a par with stipulating 3 + 5 does not equal 8.

But you are right that one ought not over stretch the metaphor. I had in mind more the almost organic way the game grows, especially how different branches develop by contradicting some of the rules.

Yes, thanks, I missed out the reply sign, that's what I meant! :up:

The second video looks at the paradox in depth and more seriously and without the teasing. You're right about the disinformation and I would think in a case like this it's probably ok in the classroom where the confusion can be sorted out afterwards - but publishing it on line may just cause more.

It looks as if @unenlightened is tiered of the game.

In this context, infinite summation is defined only for converging sequences — TonesInDeepFreeze

(converging sequences of partial sums: S(n)= a(1)+a(2)+a(3)+...+a(n) -> S, or a(n)->0 fast enough)

It's nit-picking, but there are several "summability" theories of divergent series. They assign "sums" to certain divergent series and must give the proper sum to convergent series.

Summability of Series

And Riemann showed that a conditionally convergent series can be rearranged to "sum" to any value.

In this context — TonesInDeepFreeze