'Creating' numbers.

[cheesehead]

In fact, mathematics already divides its statements for which a proof exists from its statements for which no proof is known. The latter are categorized as conjectures, or unsolved problems, ... or axioms. If proofs were not part of mathematics (so that a new proof did not constitute new mathematics), then why have distinctions between conjectures, unsolved problems, axioms, and all other mathematical statements?

Of course it could be argued that the above categories distinguish merely between statements that have no known proof and statements that have one or more known proofs. In that case, one might consider it not to be new mathematics to create a new proof of a statement for which some proof already existed, but is that desirable?

In my above example of method M, before it was proven either correct or incorrect it would have been a conjecture and thus a valid part of mathematics. After a proof of invalidity was created, then method M lost the status of (potentially valid) conjecture (for which no proof is known), and became a proven incorrect statement (another category, along with proven correct statements, to be added to ones listed above). But all that changed was a proof, not the statement/method itself.

[Numbers]

Originally Posted by cheesehead

But that implies that you think that mathematics does not include proofs of mathematical statements.

In my mind, and I appreciate that it may not have been obvious from my post, I was making a distinction between mathematics as a singular noun, and mathematics as a plural noun. So to avoid any further confusion let's use Mathematics as the plural noun for the abstract science of number, and use maths as the singular noun for the manipulation and calculation of numbers. The distinction may not always be clear and we may get confused, but hey, let's have a go.

From the distinction made above it should be obvious in which sense I meant that writing a new proof of a pre-existing fact is not inventing maths. The statement P has to actually be true before anyone can write a proof of it. Each new proof of P is a contribution to Mathematics, but makes no difference to maths. (Except in the sense that it may change methods, or teaching, etc.).

Originally Posted by cheesehead

you'll agree that mathematics has to include proofs, and therefore invention of a new proof means that new mathematics has been invented

I agree that Mathematics has to include proofs. A long time ago Euclid proved there are infinitely many primes. I recently read two other proofs of this fact. Did the writers of those proofs invent new maths (be sure to use the right definition) when they wrote those proofs? I am sure you would have to agree that they did not. They were still the same prime numbers, after all. The proofs were a contribution to Mathematics, but of negligible interest to maths, and do not (in my eyes) constitue invention.

Now let's discuss invention. As mentioned, there are at least three different proofs that there are infinitely many primes. Let's call them A, B and C. When Euclid wrote A) then B) and C) were already true, even though no one knew it. Later, someone discovered B). He did not invent B), because it was already true, he simply discovered what was already true. You might as well claim that Columbus invented America!

Originally Posted by cheesehead

But wasn't it necessary to prove that Quadratic Sieve gave correct results before it could be accepted as a valid new method?

Yes it was. The algebra that lies at the heart of a Quadratic Sieve was known for a long time, and proved by schoolboys for hundreds of years. What makes this into an invention (in my eyes) is that they took something that was already well-known, and applied it in an innovative way to (as you say) create a new method. They actually added something both to Mathematics, and to maths.

Incidentally, as an aside, the proof of the sieve was around for some time, with few followers until they actually found a big prime with it.

Originally Posted by cheesehead

Suppose I proclaim a new method M for determining whether a number is prime, but it turns out not to give correct results. Then I haven't invented any new mathematics, have I

No, you haven't. If it had worked, and if it was as innovative and creative as the Quadratic Sieve, then yes maybe you might. But as you describe it, I agree. You haven't invented anything.

Originally Posted by cheesehead

in which case there has to be a proof of its invalidity

Why does there have to a proof of its invalidity? Surely, the absence of a proof of its validity would be taken by any Mathematician to be sufficient reason not to believe it worked. After all, when amusing fellows post to this forum that they have proved x, y, and z the first thing we ask them to do is to show us their proofs. And we take the absence of proof as being proof of absence of proof. So there does not have to be a proof of your methods invalidity to prove that it does not work. (well, okay, if you need to prove that it doesn't work then I suppose a proof of that is necessary. But there is no logical reason for us to have to prove everything that doesn't work, is there?)

A proof is proof. But no proof is no proof at all.

n.b. As I am about to post this I realise you have added something else, which I have quickly glanced at but not had a chance to respond to. I don't initially think it changes anything I have said here.

[cheesehead]

Originally Posted by Numbers

In my mind, and I appreciate that it may not have been obvious from my post, I was making a distinction between mathematics as a singular noun, and mathematics as a plural noun.

Oh? Is that a British distinction? Such a difference could explain a lot.

In my Webster's Third New International Dictionary (admittedly, 1976 was a while ago), there is no distinction such as you describe. I've been assuming, as WTNID specifies, that "maths" (in a mathematical context) is simply an abbreviation of "mathematics" with no difference in meaning.

So to avoid any further confusion let's use Mathematics as the plural noun for the abstract science of number, and use maths as the singular noun for the manipulation and calculation of numbers.

Okay. I'd use "arithmetic", "calculation", "computation" or "numerical something-or-other" for the second meaning, but I can learn. :)

Why does there have to a proof of its invalidity?

Because otherwise, anyone's opinion that it's invalid carries little weight. Since a proof of invalidity could consist of a single counterexample, it's not too high a standard to require.

Surely, the absence of a proof of its validity would be taken by any Mathematician to be sufficient reason not to believe it worked.

So, you're saying that absence of proof could be sufficient grounds for an opinion. Fine.

After all, when amusing fellows post to this forum that they have proved x, y, and z the first thing we ask them to do is to show us their proofs. And we take the absence of proof as being proof of absence of proof.

That looks like a tautology. Did you mean "we take failure to show us their proof as justifying a skeptical attitude towards their claim" ?

So there does not have to be a proof of your methods invalidity to prove that it does not work.

Hunh? Are you sure you mean that? A single counterexample can prove invalidity, so I think it's reasonable to require more than just "Aw ... I don't think it works, but I can't cite any counterexample."

But there is no logical reason for us to have to prove everything that doesn't work, is there?

No, but in the absence of proof, you're just asserting opinion. There's nothing wrong with assertion of opinion, but that isn't the equivalent of proof, and proof means more than opinion in Mathematics.

[xilman]

Originally Posted by cheesehead

Oh? Is that a British distinction? Such a difference could explain a lot.

Not as far as I know.

There again, I am not the ultimate arbiter on shades of meaning within the English language and its geopgraphical and cultural variants.

The argument over the more general question of whether mathematics is invented or discovered has been raging for at least two thousand years. I see no sign of it being settled any time soon.


Paul

[Numbers]

Originally Posted by cheesehead

WTNID specifies, that "maths" (in a mathematical context) is simply an abbreviation of "mathematics" with no difference in meaning

Correct. I am simply suggesting that in this thread we adopt that convention to make the distinction clear between ourselves.

Originally Posted by cheesehead

Did you mean "we take failure to show us their proof as justifying a skeptical attitude towards their claim" ?

Yes, if you like. Perhaps I got too carried away with an amusing word construction.

Originally Posted by cheesehead

Hunh? Are you sure you mean that?

Yes. Because you forgot to mention that I did add the rider that

Originally Posted by Numbers

if you need to prove that it doesn't work then I suppose a proof of that is necessary

Yesterday I invented a new method of calculating the inverse of x modulo p. Okay it doesn't work. Are the mathematics police going to come round banging on my door demanding that I prove it doesn't work? Will the world of scientific investigation crumble under a heap of disgruntled Mathematicians if I do not write a sufficiently rigorous proof of its failure? No. The Mathematics community will simply take absence of proof that it does work as sufficient evidence that it does not. (Or quite possibly just ignore it). Every day, hundreds of Mathematicians have ideas that come to nothing. Do they labour over proofs of the fact that their idea was faulty? And where do these proofs of their non-labour get published? Which peer-reviewed journal is bursting at the seams with the proofs that bad ideas don't work? There is no logical imperative to prove that everything that doesn't work, doesn't work.

If what you appear to be claiming is in fact true, then there exists a published paper that proves you cannot run the LL test starting at both ends and meet in the middle, as was recently suggested in this forum. Cite this paper and I might believe you. And don't try and claim that a proof can exist without being published. A proof that has not been peer-reviewed ain't worth a hill o' beans. (There are exceptions to this, see below.)

Some things that don't work, and things that have previously been thought to work but which it is now known do not deserve proof of their invalidity. But not everything. Not even a significant fraction of bad ideas need a proof that they are bad ideas.

Jim and Bob meet at a conference. Over a drink Bob mentions something he's working on. On the plane home Jim thinks about what Bob said and realises why it is a bad idea. When he gets home he writes down his idea and faxes it to Bob. "Hey, Bob, here's why your idea won't work..." Bob reads it and realises that Jim is right. So now we have a proof (unpublished) that a bad idea doesn't work. Jim didn't write his proof as a serious scientific paper, he wrote it as the continuation of a Mathematical conversation with Bob. Do Bob or Jim need to publish this proof? I don't think they do, and I don't think that you would think they do either (Unless of course the idea was sufficiently Earth-shattering that a proof of the invalidity of this particular method might be of advantage to other Mathematicians working in the same field). The proof exists in the physical sense, but it does not need to exist in the scientific sense of being published. So although it is in one sense a proof, this is not the sense in which I meant proof in the whole of the rest of this thread.

So I guess that we need to prove to ourselves (each individual Mathematician needs to prove to him/herself) that our bad ideas are bad, how else would we know that they are bad ideas? But there is absolutely no good reason to publish all these proofs of why bad ideas don't work, and I am having a great deal of difficulty trying to understand why you might want to maintain that there is, unless all you are doing is playing devil's advocate rather than expressing your own views and opinions.

[Numbers]

Originally Posted by xilman

There again, I am not the ultimate arbiter on shades of meaning within the English language and its geopgraphical and cultural variants.

Perhaps not the ultimate arbiter, but an extremely well-informed and reliable one.

[mfgoode]

[QUOTE=Numbers]Correct. I am simply suggesting that in this thread we adopt that convention to make the distinction clear between ourselves.

Yes, if you like. Perhaps I got too carried away with an amusing word construction...... snip ] Unquote:/

Numbers I am not trying to discourage you in your somnambulism and verbose, copious statements where you your self are getting in-tangled.

As background reading I would advise you to take a good dose Of Plato (idea)
Russell (number) and Cantor (Sets) and then talk. I find your foundation very weak and you must build on more solid materials than sand.

Learn the masters first before you crawl so that you will be well balanced when you try walking! Its the best way to become a master yourself!

Be humble and follow those who have trudged the craggy past before you. Then only will you reach the pinnacle of achievement !
If you do, who knows you may even receive the Fields medal some day ?

BTW: What would you call ( i^i) which turns out to be a real number? Would you say its in the mind or out of it? Will it be an invention or a discovery?
Wish you all the best in your studies.
Mally