Wikipedia:Reference desk/Archives/Mathematics/2012 November 3

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November 3

More or less?

Can we say that there exists more irrationals than rationals or vice-versa, more even that odd or vice versa among other things. I think the answer is no, but if so why we say more transcendental exists than algebric and so on? 117.227.3.241 (talk) 16:40, 3 November 2012 (UTC)

I'd think you could say there are more irrationals than rationals, but should be the same number of odds and evens. For the rationals, consider just those which can be created by integers between 1 and 10 in both the numerator and denominator. That gives 100 possibilities, some of which are multiples of each other. If you then allow the numerators and denominators to be reals, we then get an infinite number of irrationals using the same ranges. StuRat (talk) 16:46, 3 November 2012 (UTC)

Infinite sets may be countable or uncountable. Rationals are countable but irrationals are not, so we usually say there are more irrationals with an implicit definition of "more". Algebraic numbers are countable but transcendentals are not, so it's the same here. See cardinality for a definition which also allows us to compare two uncountable sets. Integers and all subsets of integers are countable but other definitions of "more" can be used for integers, such as natural density. With this definition we can for example say there are more composites than primes. A more advanced definition will allow comparison of sets with natural density 0 and for example say there are more squares than cubes. It all depends on the used definition. Odds and evens will be the same in any meaningful definition where they can be compared. PrimeHunter (talk) 17:34, 3 November 2012 (UTC)

There are definitely more irrationals than rationals. PrimeHunter talks about using an "implicit" definition, which is true, but it really is the correct definition; there really, Platonistically, are indeed more of them, in the natural sense of the term. --Trovatore (talk) 08:53, 4 November 2012 (UTC)

I don't like heating the atmosphere to produce bitcoins

hi, i don't like heating the atmosphere to produce bitcoins. Please propose an alternative scheme with the same properties that does not involve proof-of-work "mining" but has the correct incentive scheme for a distributed, anonymous, peer-to-peer currency with the same properties (cryptographic and social) as bitcoin. thanks. --82.131.132.190 (talk) 22:34, 3 November 2012 (UTC)

I have just the thing, and will give you a 10% discount off of my standard $20,000 (US) consulting fee. (I do accept bitcoins.) Best, Sławomir Biały (talk) 23:07, 3 November 2012 (UTC)

I think this is an incredibly rude demand at the volunteer reference desk, and you should be ashamed. if you don't want to volunteer, then don't, but to offer consulting services to those who can't afford it is just rude. would you "volunteer" at a children's library, ask the children what they want to read, and then not only not give it to them, but try to sell it to their parents for a huge markup? shameless. 86.101.32.82 (talk) 08:57, 4 November 2012 (UTC)

I think you have overestimated the sincerity of Sławomir's offer. --Trovatore (talk) 09:35, 4 November 2012 (UTC)

Ok, fine, 25% off. But that's my final offer! Sławomir Biały (talk) 20:35, 4 November 2012 (UTC)

Before I blindly hand over money for something, I would like to have some assurance that the thing works. Since the only requirement (besides not heating the atmosphere with proof-of-works) is that it have "the same properties (cryptographic and social) as bitcoin", and you write, "(I do accept bitcoins.)", will you accept payment in the resulting currency? --91.120.48.242 (talk) 13:32, 5 November 2012 (UTC)

See Trovatore's post. I had assumed that you were trolling, but in the light of this post I am not so sure. Sławomir Biały (talk) 14:13, 5 November 2012 (UTC)

And I had assumed that where you wrote, "I have just the thing" it meant that you can produce it. Let's be serious for a moment. What would it take for you to produce an alternative to the BitCoin protocol that does not depend on "proof-of-work" 'mining' which exists literally to no other purpose than to heat the atmosphere. This is the definition of "work" in this sense. This takes one of the worse possible solutions for a monetary system - wasting resources to mine gold just to use it instead of paper - and brings it into a cryptographic digital solution where it has no place whatsoever. I would like all the benefits of bitcoin without the silly mining portion. For example, wouldn't it be possible to give coins out in exchange for "availability" - thereby incentivizing participation in the network - which can be checked statistically from time to time with simple pings, instead of for number-crunching? Other solutions are also possible. What would it take? --91.120.48.242 (talk) 14:59, 5 November 2012 (UTC)

The CPU cycles could be spent doing something useful for humanity, rather than just heating the atmosphere. Sławomir Biały (talk) 15:11, 5 November 2012 (UTC)

That would be acceptable to me as long as you don't produce a "nonce" need (i.e. that you just happen to discover). Also I hold a level of incredulousness toward some distributed (e.g. Seti@home or finding twin primes or something) projects, I think they are a waste of resources. So you would have to make a pretty good case for the work. Do you have any other solutions or suggestions? (To be sure, what you've just said is a good, acceptable solution.) The idea of *making* someone do work to produce the scarcity is one that I feel is cryptologically unsound. There should be something better. --91.120.48.242 (talk) 15:37, 5 November 2012 (UTC)

As I already said, the "creation", or more accurately the initial distribution, of the currency is only temporary and insignificant long-term. The real purpose of hashing is to synchronize transactions. Maybe a better way to decide the initial distribution is possible but that's irrelevant if you can't find a better way to synchronize transaction.

As I already said, the proof-of-work system isn't some thing slapped on to Bitcoin and which needs to be removed for Bitcoin to achieve perfection. The proof-of-work system is Bitcoin and there is no other known way to do what Bitcoin does.

You'd be taken more seriously if you try to reasonably debate the problem and proposed solutions (such as the proof of stake already mentioned). -- Meni Rosenfeld (talk) 18:34, 8 November 2012 (UTC)

Instead of backing your bitcoins with precious metals, you can back them with property. Since land values are low now, this would be a good time to buy it up and parcel it out, no mining required. StuRat (talk) 23:21, 3 November 2012 (UTC)

The proof-of-work system in Bitcoin serves two purposes:

1. Determine the initial distribution of the bitcoins.
2. Synchronize transactions to prevent double-spending.

The reason Bitcoin didn't exist before 2009 is that prior to Satoshi Nakamoto's invention of the Blockchain, there was no known way to do #2 in a decentralized way (and the known ways to do #1 without being prone to Sybil attacks also involved proof of work). Thus there is no known way to operate a decentralized digital currency without proof of work, and developing one, even if such a thing is possible, is way beyond the scope of this reference desk.

By your own admission, having a cryptographic peer-to-peer currency is extremely important, and the energy expenditure required to maintain it is a small price to pay (arguably, smaller than the resource waste of maintaining the current monetary system).

Purpose #1 is only temporary, the long-term purpose is #2. And there are hypothetical proposals for how to prevent double-spending while minimizing energy consumption, using some form of proof of stake. There is also an alternative digital currency called PPCoin which uses related ideas.

See also https://en.bitcoin.it/wiki/Myths#Bitcoin_mining_is_a_waste_of_energy_and_harmful_for_ecology.

FWIW, I'll be away for a week and unable to follow up on this discussion. -- Meni Rosenfeld (talk) 04:53, 4 November 2012 (UTC)

I can't help wondering whether it is possible to do some useful work like "SETI@home" or "grid.org" and at the same time have a probability of computing a bit-coin like currency, possibly out of "misses" on the main project with some peculiar mathematical property. It feels like it should be possible and if it were it would be a "win-win", with energy used for currency mining having a positive side effect and more people working on these useful problems -- Q Chris (talk) 09:46, 6 November 2012 (UTC)

I doubt it. Bitcoin PoW crucially depends on the following properties:

1. It is easy to verify that work has been done.
2. It is easy to unambiguously determine the difficulty of the work.
3. It is possible to create, in a decentralized way, a problem which references arbitrary data so that any change in the input data renders the solution completely invalid. (Otherwise it is impossible to use the work to confirm a given set of transactions.)

Correspondingly, the main challenges with adapting "useful" work for synchronizing transactions (and probably also initial distribution) are:

1. Many problems are as hard to verify as they are to solve.
2. Many problems don't have a clean measure of difficulty. Even if some heuristic is used to estimate difficulty, the system could be "gamed" by choosing problems which appear more difficult than they are. And most importantly,
3. Useful problems can't generally be custom-made for arbitrary data. Particular instances are identified by some central domain expert.

-- Meni Rosenfeld (talk) 18:34, 8 November 2012 (UTC)