The sum of any two integers is zero.

So much for that.

The product of any two integers is omega. (Where omega is the first number bigger than any integers).

No two rules can be combined and none can be used more than once.

Banno subtracts 50% of his dough and adds it to my dough. It's a rule. Give up your dough, Banno.

Here's a game about the philosophy of mathematics. — Banno

Math rules are discovered, not made.

Zero is the total product of one's life.
You start with it and anything added is left behind at the end.

Math rules are discovered, not made. — frank

If it is going to be discovered, then it is covered...

And hence, it is.

Where are mathematical expressions before they are discovered?

The simple answer - they are not discovered.

Then integers takes on a use that is peculiar to this game.

So much for that. — creativesoul

That shows a lack of imagination.

This is inconsistent with

The sum of any two integers is zero. — John Gill

So, do you choose inconsistency, or reject John Gill's formulation?

I don't bake.

I didn’t realize I was creating an inconsistency, so I leave it up to another poster (first comer) to decide to reject either my axiom or John’s.

That's a good move.

There may be a way around the inconsistency.

So we have, from @John Gill,

For any two integers a,b, a+b=0

And we have from you,

For any two integers a,b, ab = Ω

(The italics indicate the special in-game nature of the word 'integer').

Now ab is just a+a+a..., b times. But substituting a for b in John's rule, a+a=0

Hence, skipping a bit, 0=Ω.

And hence, a+b = Ω

We might treat this as a definition of integer, such that an integer is any number that, added to another integer, yields Ω.

The thread had been moved into the lounge. I've moved this back into Philosophy of Mathematics.

It's credentials as such should be evident in the content.

Also, extending the game is dependent on creativity - and hence on a large number of folk participating.

The contention here is that this game has similarities to mathematics, in that the playful creation of rules is at the core of both.

Since it is clear that this game is constructed, not discovered, the game is a rejection by example of the doctrine set out here:

Math rules are discovered, not made. — frank

it is going to be discovered, then it is covered...

And hence, it is.

Where are mathematical expressions before they are discovered?

The simple answer - they are not discovered. — Banno

My rule is that they are discovered. Don't toss away my rule because of some made-up logic.

Here's a game about the philosophy of mathematics.

Players take turns to add rules.

Your turn. — Banno

I add that there exists Absolute Infinity.

Is then Ω absolute infinity or not?

(I like Banno's game :up: )

Don't toss away my rule because of some made-up logic. — frank

I'm tossing it away because it is silly.

well,

Where omega is the first number bigger than any integers — Pfhorrest

hence we might presume a number bigger than the first number bigger than any integer.

But can there be an integer bigger than another integer?

What does bigger look like here?

Well, what do you do so you can give half of the fruits of your labor to me?

Math games with arbitrary rules are a useful waste of time. If you really want rules, reality has some for you. For math or language to really be of any use, they need to inform and predict the world as it was, is and will be.

We choose when and where to apply the rules.

Let's call them Gill integers.

A GIll Integer differs from other integers in that when summed, they add to zero.

Now, is there more than one Gill Integer?

Let's call them Fhorrest Integers.

Are they the same as Gill integers?

"Rules" is probably the wrong term to use. Any mathematics without real-world applications would be the game you're looking for. Knowing how many miles to the next rest stop and how fast you are going isn't a game when you really need to empty your bladder. It produces true knowledge about you and the world. Is your game useful for anything outside of this thread?

I'm tossing it away because it is silly. — Banno

No. You're tossing it away for no reason.

The Axiom of No Choice: For any collection of non-empty sets there is at least one way to avoid choosing an element from each set.

(This will lead to a pathological nightmare in the case of an uncountable infinity of such sets)

The Axiom of Inclusion: Given two empty sets, one is the absence of an element of the other.

:nerd:

That shows a lack of imagination. — Banno

Me???

:lol:

Surely you jest. Maths are beyond my understanding. I don't want to be a bullshitter!

:wink:

I'll watch. Have fun.

again, what does this lead to? Is it fun to play?

Some rules lead to a more interesting game.

The preference for consistency is one such rule.

Let's say the rules of arithmetic are arbitrarily made up, like Banno's math game. The golden ratio is one result of arithmetic. The surprising thing is that it can be find in spiral patterns in nature. Now why might that be? Perhaps the rules or arithmetic are not so arbitrary.

Let's go back to their origins. How did humans come up with arithmetic? Probably when it became useful to track transactions and taxation. And that's not arbitrary.