Mathematics

[Revoltingest]

If you take 2 quarts of water and add 2 quarts of ethyl alcohol (grain alcohol), you will NOT get 4 quarts of mixture, but a bit less.

Once again, whether the formal conclusion that 2+2=4 applies is a matter of observation and testing. Sometimes it does. And sometimes it doesn't.

This isn't so much about math, as the fact that mathematical models of a material phenomenon can be erroneous.

 

[Polymath257]

List of unsolved problems in mathematics - Wikipedia

I'd say there are more than a hundred unsolved problems listed in that article. If mathematics were merely a human invention, then there wouldn't be such problems--we would just invent the answer. Instead, mathematicians spend endless hours trying to discover what the correct answers are.

Once again, my position is that we invent the rules and we discover their conclusions. To solve these problems means to do so in the context of certain rules we invented.

To choose different rules means you are playing a different game. And different games *do* exist. But, for the most part, mathematicians have agreed to the basic rules and want solutions to problems that stay within those rules.

 

[Polymath257]

This isn't so much about math, as the fact that mathematical models of a material phenomenon can be erroneous.

Which means the math is not universally applicable. Which is what I stated.

The math consists of some formal rules: a game we play. Sometimes, and to our great advantage, those rules *also* apply to the real world.

 

[Revoltingest]

Which means the math is not universally applicable. Which is what I stated.

I'd say math is universally applicable.
But humans reserve the right to get it wrong.
One cannot simplistically apply arithmetic to the real world.
Example.
10 crows sit on a wire.
I shoot one.
How many are left?

Spoiler: Answer

None.
Firing the gun scared all the survivors away.

 

[Revoltingest]

Another couple math problems.....

Problem #1
-------------------------------------
I just dug a hole which is
23 inches wide
7 feet long
18" deep

How much dirt is in it?
Don't peek til you have an answer!

Spoiler: Answer

None

Problem #2
-----------------------------------------------------
Which weighs more....
One pound of gold
One pound of feathers

Spoiler: Answer

A pound of feathers
A troy pound has only 12 ounces.

Problem #3
-----------------------------------------------------
Which weighs more....
One ounce of gold
One ounce of feathers

Spoiler: Answer

An ounce of gold
A troy ounce is heavier than an avoirdupois ounce.

 

[Curious George]

Not at all what happens in rings. The symbol 1 is *only* used as a place holder for the thing that satisfies 1*x=x for all x.

.

How exactly is the concept of whole not assumed in any ring as well.

I am saying that the concept of whole must be assumed for any math.

Now did we invent this concept of whole does it exist external to us?

 

[Nous]

Once again, my position is that we invent the rules and we discover their conclusions.

Then why do these dozens of unsolved problems exist? No one knows what rule was invented?

The idea that mathematics is merely a human invention is a wonderful example of formulating answers to questions on the basis of one's metaphysical beliefs. Metaphysics first.

 

[Polymath257]

Then why do these dozens of unsolved problems exist? No one knows what rule was invented?

Because we agreed upon the rules of the game: the basic axioms of mathematics as given in, say, Zormelo-Frankl set theory.

The *goal* is to answer those questions within that axiom system.

The idea that mathematics is merely a human invention is a wonderful example of formulating answers to questions on the basis of one's metaphysical beliefs. Metaphysics first.

Like I said, the rules are invented (like those of chess) and the results are discovered (like the number of moves to checkmate).

We choose the rules to agree with some of our intuitions. But we also know that many of our intuitions are self-contradictory.

 

[Polymath257]

How exactly is the concept of whole not assumed in any ring as well.

I am saying that the concept of whole must be assumed for any math.

Now did we invent this concept of whole does it exist external to us?

Interesting. The 'concept of the whole' has never appeared in any math class I have ever taken, nor any math book I have seen (unless you consider the class of all sets to be a 'whole').

 

[Curious George]

Interesting. The 'concept of the whole' has never appeared in any math class I have ever taken, nor any math book I have seen (unless you consider the class of all sets to be a 'whole').

Really?

Are you sure? What is a whole?

 

[Polymath257]

Really?

Are you sure? What is a whole?

A whole what?

It is seriously not a concept that is used in mathematics.

 

[Curious George]

A whole what?

It is seriously not a concept that is used in mathematics.

So, perhaps you are saying this because you don't understand a concept of a whole?

 

[Nous]

Because we agreed upon the rules of the game: the basic axioms of mathematics as given in, say, Zormelo-Frankl set theory.

The *goal* is to answer those questions within that axiom system.



Like I said, the rules are invented (like those of chess) and the results are discovered (like the number of moves to checkmate).

We choose the rules to agree with some of our intuitions. But we also know that many of our intuitions are self-contradictory.

Prove that your claims are true.

Obviously Pythagoras did not simply invent the rule for right triangles in which a2 + b2 = c2.. He discovered and proved that rule. There is no other rule that is correct for the sides of right triangles on a plane. a2 + c2 = b3 is never correct for right triangles on a plane.

Why don't mathematicians simply invent a rule to solve Hodge's conjecture? Provide the citation where someone invented the rules that lead to that conjecture.

 

[LukeS]

Does a falling tree make a sound when there no one to listen to it? Well there is a sound when we are there, for sure. Did we invent or discover this sound?

 

[Polymath257]

So, perhaps you are saying this because you don't understand a concept of a whole?

If we are working inside a set X, then the 'whole' is the set X.

Other than that, the concept isn't one used in mathematics.

 

[Polymath257]

Prove that your claims are true.

Obviously Pythagoras did not simply invent the rule for right triangles in which a2 + b2 = c2.. He discovered and proved that rule. There is no other rule that is correct for the sides of right triangles on a plane. a2 + c2 = b3 is never correct for right triangles on a plane.

You have the answer is what you wrote: he was working under the rules of plane geometry. Those are very specific rules including the axioms set down by Euclid and alayzed long before him.

Why don't mathematicians simply invent a rule to solve Hodge's conjecture? Provide the citation where someone invented the rules that lead to that conjecture.

Mathematicians work, usually, under the axioms of Zormelo-Franklset theory (usually with the axiom of choice added in). Those are the rules that were invented. To even formulate Hodge's conjecture requires the rules of that set theory. The *consequences* of those rules are discovered. We want to see if Hodge's conjecture follows from those specific rules.

Your question is sort of like asking why we can't change the rules for a chess problem and use some from checkers. To do so would be a different game.

 

[Polymath257]

Prove that your claims are true.

Obviously Pythagoras did not simply invent the rule for right triangles in which a2 + b2 = c2.. He discovered and proved that rule. There is no other rule that is correct for the sides of right triangles on a plane. a2 + c2 = b3 is never correct for right triangles on a plane.

I'd point out that with even very small changes in the rules (of plane geometry), the Pythagorean relationship becomes false. So, for example, in spherical geometry, it is *never* true.

 

[jonathan180iq]

Both, I think...
Math itself is mostly invented.
The relationship between things seems to be innate - but that could just be a product of the limitations of our senses and our ability to comprehend...
That being said, until it is proven otherwise, we have to assume that those relationships are unchanging, at least at the macroscopic layer.

 

[jonathan180iq]

Your question is sort of like asking why we can't change the rules for a chess problem and use some from checkers. To do so would be a different game.

THIS.

 

[Nous]

You have the answer is what you wrote: he was working under the rules of plane geometry. Those are very specific rules including the axioms set down by Euclid and alayzed long before him.

Euclid didn't claim to have invented any rules of plane geometry.

If your claims were true, you would be able to provide the citations showing that someone invented the rules of plane geometry, and you would be able to show that those rules are arbitrary, invented, not discovered.

Mathematicians work, usually, under the axioms of Zormelo-Franklset theory (usually with the axiom of choice added in). Those are the rules that were invented. To even formulate Hodge's conjecture requires the rules of that set theory. The *consequences* of those rules are discovered. We want to see if Hodge's conjecture follows from those specific rules.

Your question is sort of like asking why we can't change the rules for a chess problem and use some from checkers. To do so would be a different game.

No, my question is not like asking why we can't change the rules for a chess problem and use some from checkers." There are no unsolved problems in chess. There never were any unsolved problems in chess. And one can change those rules. As kids, my brother and I used to play Monopoly with changed rules. It worked out find. It didn't lead to any unsolved problems.