Mathematics

[Polymath257]

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.

The long history of the parallel postulate and the existence of non-Euclidean geometry shows this. The rules were chosen to obtain certain results. There are other possible rule sets that obtain different results.

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.

Of course there are unsolved problems in chess!!! For example: is there a winning strategy for chess?

 

[Revoltingest]

The long history of the parallel postulate and the existence of non-Euclidean geometry shows this. The rules were chosen to obtain certain results. There are other possible rule sets that obtain different results.

As I see it, the postulates are both discovered & invented.

Of course there are unsolved problems in chess!!! For example: is there a winning strategy for chess?

It is not a "solved" game....yet (perhaps).

 

[Polymath257]

As I see it, the postulates are both discovered & invented.

And there is an extent to which I agree.

Usually, we choose axioms to agree with certain intuitions. So, for example, prior to Euclid, we formed certain intuitions about geometry and even had proofs linking various geometrical concepts.

But it was Euclid (as far as we know) that invented the specific set of axioms that became the standard definition of plane geometry. He certainly chose those axioms because they allowed the proof of results that were previously seen as important for our intuitions for plane geometry. So he discovered a set of axioms that conformed to his intuitions.

But we now know that those axioms were in no way 'required' nor are the results that follow from them. For example, there are non-Euclidean geometries where the sum of the angles for any triangle is always *less* than 180 degrees. Where the degree of 'deficiency' from 180 degrees is actually proportional to the area of the triangle.

And this is a perfectly consistent geometry. It just has different results from ordinary Euclidean geometry.

It is not a "solved" game....yet (perhaps).

Precisely. There are problems that are unsolved. Just like in mathematics. To solve the Hodge's conjecture requires a proof one way or the other: which is a sequence of statements (moves) that goes from known results (starting position) to the result desired (ending position). The analogy is actually quite accurate.

 

[Nous]

The long history of the parallel postulate and the existence of non-Euclidean geometry shows this. The rules were chosen to obtain certain results. There are other possible rule sets that obtain different results.

So your claims of people having invented the rules of mathematics are false. You can't provide a single citation of anyone claiming to have perpetrated such an invention.

Of course there are unsolved problems in chess!!! For example: is there a winning strategy for chess?

That isn't an unsolved problem in chess. People employ winning strategies in chess every day.

 

[Polymath257]

So your claims of people having invented the rules of mathematics are false. You can't provide a single citation of anyone claiming to have perpetrated such an invention.

Yes, I just did. Euclid invented one system. Labochevsky invented another system for geometry. Zormelo invented one for set theory.

That isn't an unsolved problem in chess. People employ winning strategies in chess every day.

You clearly don't understand the question. Is there a winning strategy, for which a perfect player would be able to guarantee a win from the starting positions?

 

[blü 2]

So p1 x p2 = p3 x p4 ?

I'd have said that the number produced by multiplying primes together using each discrete prime only once had unique factors.

Since they're unique, no other selection of primes each used once could produce the same result.

Now I think further, that seems to be true of primes even if you use them more than once, as long as core set of primes A is not the same as core set of primes B.
.

 

[blü 2]

You can certainly invent very many, perhaps infinitely many, kinds of maths just by specifying the rules.

Cantor (inter alia) invented the "lowest infinite ordinal" ω. The idea that there are parts of the number line that can't be reached because they're soooo distant from the origin is ─ and I'm being polite here ─ not intuitive. Still, people seem happy to bounce the resulting maths around. I've never seen them used for anything except a few puzzles like the Hotel Infinity, but that, of course, isn't the test.

 

[Polymath257]

So p1 x p2 = p3 x p4 ?

I'd have said that the number produced by multiplying primes together using each discrete prime only once had unique factors.

Since they're unique, no other selection of primes each used once could produce the same result.

Now I think further, that seems to be true of primes even if you use them more than once, as long as core set of primes A is not the same as core set of primes B.
.

You can have multiple primes on each side and repetitions. The question is then whether both sides have to have the same primes and the same number of each prime.

So, for example, 444=2*2*3*37=3*2*37*2=37*2*3*2. All use the same primes and the same number of each, just re-arranged.

 

[Polymath257]

Typical math student:

Saturday Morning Breakfast Cereal - A New Method

 

[blü 2]

You can have multiple primes on each side and repetitions. The question is then whether both sides have to have the same primes and the same number of each prime.

Yes, they do.

For example 28 = 2 x 2 x 7 (2 x 7 x 2, 7 x 2 x 2). Full stop. No other primes than 2, 2 and 7 will get you to 28. And in general any integer n has a unique set of prime factors.
.

 

[LukeS]

Axioms are invented, but their implications are discovered? But, aren't axioms meant to be intuitively true...?

 

[Polymath257]

Axioms are invented, but their implications are discovered? But, aren't axioms meant to be intuitively true...?

Well, that was the original intent. But we have found that this isn't such a good guide. The 'intuitive' set theory is self-contradictory. Euclid left out several important axioms on 'betweenness'. And whether the parallel postulate is 'intuitively true' was debated for a couple thousand years before a completely different geometry was discovered where it is false.

So, yes, intuition guides the choice of axioms. We have ideas that we want to model in our language of mathematics and we choose axioms to do those models. But there is NOTHING that uniquely chooses any one set of axioms or even one axiom system (which could have different, but ultimately equivalent axioms).

One of the things we have found over the past 200 years or so is that there are many alternative axiom systems (for example, Labochevskian geometry) that are equally consistent, but just give different results. A big part of math these days is investigating the different alternatives and seeing where they lead.

In fact, each branch of mathematics and even each subdivision of each branch has its own collection of axioms, although often with the axioms of set theory as a base.

 

[Polymath257]

Yes, they do.

For example 28 = 2 x 2 x 7 (2 x 7 x 2, 7 x 2 x 2). Full stop. No other primes than 2, 2 and 7 will get you to 28. And in general any integer n has a unique set of prime factors.
.

Yes. And, in fact, unique factorization in this sense is true for integers. But I would say that it is far from 'obvious', especially when you go beyond 'small' numbers.