Contradiction!

Proof:

The sum of any two integers is zero. — jgill

No two rules can be combined and none can be used more than once. — I like sushi

Rule 1: The sum of any two integers is 0. — Lionino

Every contradiction shall be resolved both ways.

Then surely the sum of any two integers is 0, and we must accept that two rules can be combined and that all can be used more than once

Or we must never reference a previous rule to even your post, and the sum of any two integers is the sum as we understand it from the textbooks.

And, having said this, the first is the assertion, the second the negation, and now I'm wondering -- what's the negation of the negation?

Rule 1: The sum of any two integers is 0 — Moliere

That was me referencing jgill, not making up another rule :^)

It's the 'many worlds' interpretation of mathematics.

Always "and", and never "or", but also "or"... Etc.

Banno's thesis is that maths is invented, not discovered, just as games like chess are. Well then it is very easy to invent some rules for a game or some rules for a mathematics, and there are lots of them. But most are dull or unplayable.
So the thread itself is badly set up as a game that doesn't have much interest or significance, because posters can, and nearly always do, take the nuclear option and pretend they have "won". A better win might be if we could come up with a new form that was consistent and incomplete, but not isomorphic with arithmetic or something like that. I don't have a better set up that would encourage that, unfortunately.

A better win might be if we could come up with a new form that was consistent and incomplete, but not isomorphic with arithmetic or something like that. I don't have a better set up that would encourage that, unfortunately. — unenlightened

Me either.

Though I think your insight here is worth preserving:

So the thread itself is badly set up as a game that doesn't have much interest or significance, because posters can, and nearly always do, take the nuclear option and pretend they have "won" — unenlightened

The nuclear option -- contradiction -- is something like the fruit on the tree in paradise?

Banno's thesis is that maths is invented, not discovered, just as games like chess are. Well then it is very easy to invent some rules for a game or some rules for a mathematics, and there are lots of them. But most are dull or unplayable. — unenlightened

This, though, is the stronger point.

If the King is in check then the other player can swipe away the peices, but this is rude (and so it goes with the other games; the dull and unplayable games seem to proliferate, and the interesting ones are the ones we ought go for)

I think math is probably like chess, but that chess was built upon mathematics: so the metaphor is good, but starts on the wrong side.

Derivative problem. If you are a platonist, you think math is discovered, if you are a nominalist or conceptualist, you think math is invented.

So the thread itself is badly set up as a game that doesn't have much interest or significance — unenlightened

And yet it lives, five years on.

If the King is in check then the other player can swipe away the peices, but this is rude — Moliere

Some rules ruin the game, others make it more interesting.

One way to fix the game might be to oblige players to list the rules they are making use of, and hence have them construct a tree.

Hence,

Players take turns to add rules. — Banno

The sum of any two integers is zero. — jgill

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

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

Conclusion:

0=Ω — Banno

Let's call them Gill integers. — Banno

Let's call them Fhorrest Integers. — Banno

Question: prove that Fhorrest integers are the same as Gill integers

Theorem 1: Any two integers are the opposite of each other
a=-b — Lionino

(from JGill's rule)
Conclusion:

There is only one integer, 0. — Lionino

An adding: If there is only one integer, then Fhorrest integers are the same as Gill integers.

New rule: There is an integer that is neither a Fhorrest integers nor a Gill integer.

Your turn...

If the King is in check then the other player can swipe away the peices, — Moliere

That might be (but actually isn't) an interesting game, but it is no longer chess. Allegedly, rugby was invented when some idiot was supposedly playing football and picked the ball up and ran with it. A few other things had to change before it became a game worth playing.

There is a card game called "52 card pick up", in which the dealer throws all the cards up in the air, and leaves their opponent to pick them up. It's faintly amusing. Once.

And yet it lives, five years on. — Banno

As does 52 card pick up. But if you want to do something interesting in mathematics, or the philosophy of mathematics, this is not the way to go about it.

But if you want to do something interesting in mathematics, or the philosophy of mathematics, this is not the way to go about it. — unenlightened

But yet again, here you are…. :wink:

Don't measure your success by my presence; I am a notorious shoveler of shit.

Well, I'm only too pleased to provide you with the raw material.

New rule: There is an integer that is neither a Fhorrest integers nor a Gill integer. — Banno

I think that is contradictory with Fhorrest's rule.

Derivative problem. If you are a platonist, you think math is invented, if you are a nominalist or conceptualist, you think math is discovered. — Lionino

LOL

I messed that up. It is the inverse, fixed now.

Wow I had no idea that my offhand contribution here would still have such a lingering impact so many years later...

(FWIW though, my name isn't P. Fhorrest, it's just Forrest but spelled with a Pfh instead of an F).

Man perishes; his corpse turns to dust; all his relatives return to the earth. But writings make him remembered in the mouth of the reader. A book is more effective than a well-built house or a tomb-chapel, better than an established villa or a stela in the temple! — Ancient Egypt, 12th century BC

That might be (but actually isn't) an interesting game, but it is no longer chess. Allegedly, rugby was invented when some idiot was supposedly playing football and picked the ball up and ran with it. A few other things had to change before it became a game worth playing.

There is a card game called "52 card pick up", in which the dealer throws all the cards up in the air, and leaves their opponent to pick them up. It's faintly amusing. Once. — unenlightened

I'll admit that that's not my favorite game. And there are only so many times I can play it.

Though what becomes shit was at one point food

Derivative problem. If you are a platonist, you think math is discovered, if you are a nominalist or conceptualist, you think math is invented. — Lionino

Hrm. What's it derived from? "How does math work?" ?

Derivative problem. If you are a platonist, you think math is invented, if you are a nominalist or conceptualist, you think math is discovered. — Lionino

What of us who think it is both created and discovered?

Nice to read a new post from you, Forrest.  :smile:

New rule: There is an integer that is neither a Fhorrest integers nor a Gill integer.

The sum of any two integers is zero. — jgill

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

It must be a number smaller than omega, but not zero.

Something like this: x (the suspicious integer) < Ω.

It is 1. Why? Because it is the smallest integer greater than zero and the smallest of Omega.

Everything I wrote above is pure crank, right? :lol:

What's it derived from? — Moliere

From what mathematical objects are.

What of us who think it is both created and discovered? — jgill

Sounds contradictory to me, unless you are saying the application of it is discovered.

What of us who think it is both created and discovered? — jgill

Sounds contradictory to me, unless you are saying the application of it is discovered — Lionino

It is a bit fuzzy. But here is an example: Linear fractional transformations have been around for years but at some point someone discovered they could be categorized by the behavior of their fixed points. One such categorization was "parabolic", in which the fixed point demonstrates both attracting and repelling behaviors. Thus, a category was both discovered and created. When I determined the conditions under which infinite compositions of parabolic transformations converge to their fixed points years ago that was a discovery based upon a creation.

And speaking of which, category theory could be considered a creation, then its characteristics follow as discoveries.

However, I am open to other perspectives. Most mathematicians don't care to argue the point. But it is certainly fair game for the philosophically inclined.

When I discovered the conditions under which infinite compositions of parabolic transformations converge to their fixed points that was a discovery based upon a creation. — jgill

Though I am completely out of my depth when it comes to the example you brought up, it seems that is a case of one discovering a consequence of one's invention — the consequence was invented, just unintentionally.

But "discovery" and "invention" can be sifted by asking if the system in question would still be true had humans never put in paper. If so, discovery; if not, invention.

The square root of 2 is now 1.5. Nice and easy lol

Been a long time buddy. Come back to the forum! It needs all the rationality it can get at this moment in history :lol:

Thinking a bit more about this, it does seem like discovering a category or theorem of a theory or its objects spells some trouble to someone who thinks math is purely invented, but under some conditions. For conceptualists, who think that mathematics is invented by the human mind, it is not clear how one could derive a theorem from existing facts without referencing something that was already set in place, independent of a mind — with that reference, it is no longer purely a mental activitity. Though they could make up arguments, like that mathematical truths are necessary from the axioms, but then their distinction from formalism/nominalism is not so clear cut. Even then, one would wonder, what is it that makes the fact necessarily follow from the axioms? Perhaps the laws of logic, but then we are back to Russell trying to reduce the foundations of mathematics to logical statements, and then Gödel...

You're taking the game too seriously. But it is the attitude I've seen most people take in posts discussing logic, riddles, games, etc. When they interact with me I feel hugely overwhelmed.
They believe I am trolling, but I simply lack wording and reasoning.

New rule: The sum of the product of any two integers is omega minus (now corrected!) the double of Lionino’s integers. 

I was playing the game, but people brought up philosophy of mathematics so I took the chance to talk about it — better than the pointless merry-go-rounds about ethics we often have here.

Because you asked so nicely and I can't help myself, I'll chime in briefly on the topic of whether mathematics is created or discovered:

I think that the distinction between creation and discovery only applies to concrete things, and for abstract things like mathematics there is no such distinction. Because there we're dealing entirely with matters of possibility, so to discover something is just to show that it is possible, as in, it could be created, at any time; and conversely, to create something is only to show, and so discover, that it is possible, and always has been.

It's only with concrete things that exist within time that they could have already been actualized (in the past) and so be available to be discovered, in a way distinct from not having been actualized yet and so being available to create (in the future).

Rule:
Vector spaces may have irrational dimensions.