Conclusion:
The main idea for those virtual subset is to enlarge the set theory.
In general it is neither decidable if V has in minimum one element nor if V have finite
numbers of elements or infinite many.
That leads to the possibility of a virtual set having finite elements but this elements are not
countable. Those finite but not countable virtual subsets will be a new category of the set
theory.

He came to this conclusion with reference to Gödel and undecidabilty: you don’t get better mathsiness than that!

I also came across DEGREE OF NEGATION OF EUCLID’S FIFTH POSTULATE (ALLCAPS in the original) by Florentin Smarandache, a paper misfiled as GM by the INTERNATIONAL MAFIA IN SCIENCE. Smarandache is an associate professor of mathematics at the University of New Mexico. Clearly he is rather distinguished, as he has a entry on PlanetMath and a Wikipedia entry with a large number of other articles linking to it all of which where, no doubt, written by completely neutral students of his work and describe highly notable phenomena. Also there is a journal committed solely to discussion of his ideas. Also.

He gives us this.

In this article we present the two classical negations of Euclid’s Fifth Postulate (done by Lobachevski-Bolyai-Gauss, and respectively by Riemann), and in addition of these we propose a partial negation (or a degree of negation) of an axiom in geometry.
The most important contribution of this article is the introduction of the degree of negation (or partial negation) of an axiom and, more general, of a scientific or humanistic proposition (theorem, lemma, etc.) in any field – which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood), or like in neutrosophic logic [with a degree of truth, a degree of falsehood, and a degree of neutrality (i.e. neither truth nor falsehood, but ambiguous, unknown, indeterminate)].

It’s good that it works with “a scientific or humanistic proposition…in any field”? Applicability and interdisciplinary work go down well with funding bodies (at least the non-MAFIA controlled ones). Sadly this particular paper is rather light on details

Also notable is that he developed his theory of geometries which are sometime Euclidean and sometimes not because he “observed that
in practice the spaces are not pure, homogeneous, but a mixture of different structures.” I’ve noticed before a tendency amongst less narrow-minded thinkers in mathematics to believe in a very special kind of Platonism, where there exist really real mathematically entities but they aren’t things other mathematicians study. Only the thinker’s own concepts relate to really real things.

And no, I never did say TWFiGM would be every week.

The Streets, Everything is Borrowed. Mike Skinner seems to walk a narrow line with “fantastic” on one side and “ridiculous” on the other. This is the former.

So anyway, maths undergrads reading this, you can do induction. Well done. Good. Let’s do induction on strings of the word “buffalo”.

The single best title of any article on Wikipedia is Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo
. The notable thing about this string of “buffalo” is that it can be parsed as a sentence. This fact was first noted by the computational linguist William Rapaport of (naturally) the University of Buffalo. We use the facts that:
“buffalo” is a plural noun (1 buffalo, 2 buffalo,…);
“to buffalo” is a (somewhat rare) verb meaning “to bully”;
“Buffalo” is a city in the American state of New York and in English we may use city names as adjectives (e.g. “the London mayor is a laughing stock”).

This lets us parse the sentence as follows (this is taken form Wikipedia).

[Those] (Buffalo buffalo) [whom] (Buffalo buffalo buffalo) buffalo (Buffalo buffalo).

or

THE buffalo FROM Buffalo WHO ARE buffaloed BY buffalo FROM Buffalo ALSO buffalo THE buffalo FROM Buffalo.

. OK, but what does that have to do with proof by induction? Well we claim that, for each natural number n, we can repeat the word “buffalo” n times and then (give or take capitalisation) parse it as a sentence**.

The easiest way is to use induction to show that the odd cases (i.e. when there is k with n=2k+1) we can parse the sentence without needing the city meaning of “Buffalo”.

We have to treat n=1 as a special case, so lets take n=3 as our base case. We have the sentence,

Buffalo buffalo buffalo.

That is,

Bison bully bison.

Now we can add two “buffalo” by stating that the first buffalo themselves get buffaloed. That is

Buffalo buffalo buffalo buffalo buffalo.

read as

Buffalo [whom] buffalo buffalo [also] buffalo buffalo.

We can continue from there by pushing our first buffalo further still down the pecking order.

Buffalo [whom] (buffalo [whom] buffalo [also] buffalo) buffalo [also] buffalo buffalo.

Now we see what our induction ought to be. At each stage we have a sentence about a complex hierarchy of ungulate aggression,

Buffalo [whom] (buffalo [whom] (buffalo [whom](buffalo[whom].. … (buffalo [whom] buffalo [also] buffalo) … [also] buffalo)[also] buffalo)[also] buffalo)[also] buffalo buffalo.

We can obtain a sentence two “buffalo” longer, and of the same form, by adding that the buffalo at the top of current description are buffaloed in turn by still more dominant buffalo; that is by replacing the noun “buffalo” in the middle of the many nested brackets (the one with “[whom]” before it and “[also]” after it) with the noun phrase “buffalo [whom] buffalo [also] buffalo” By induction all odd length sequences of “buffalo” are sentences.

Finally we can add in a “Buffalo” (geographical adjective) to get the even length case. Note this isn’t actually how it is done in Rapaport’s original sentence where all buffalo are Buffalo buffalo.

It’s left as an exercise for the reader to write a maths post based on exploding whale

*Non-mathematicians should note the difference between mathematical induction and inductive reasoning. In maths speak induction is deductive.
**You may wish to argue as to whether that “Buffalo!” (an exclamation or order) is a sentence, in which case I say “meh” to your pettifogging pettifoggery.

Anyway the “proper” understandings of what the holiday is held to mark always left me bit cold but Woody’s version connects for me.

matt heath, marcus du sautoy gay, (s + c) x (b + f)/t – v, where s = overall shape (“including tendency to droop”), c = circularity, b = bounce factor (not to be confused with “wobble”), f = firmness (with perfect being “like a comfy bed”), t = skin texture and v = vertical ratio (the goal: “on the top-heavy side of symmetrical”). for the male rear end, the equation replaces bounce, circularity and vertical ratio with m (muscularity), l (leanness) and o (overall symmetry), listen number, immigrant expatriate

This amused me slightly. Of course now I’ll attract people searching for these even more.

I hope the project works. The sort of information that it is designed to contain is not easily available anywhere else. It’s the nature of maths that you can give a completely convincing explanation of why something is true (a proof) without giving away very much at all about how you came to know it was true (the endless headdesking before the proof). AFAIK there is no other field of study where this is true to anything like the same extent. For better or worse, the normal style of a maths journal paper positively encourages hiding the process that lead you to the result. Thus, the only way you can usually find out tools for proof is by having somebody show you, and it’s obviously rather hit-and-miss as to whether you will ever explain your problem to the person with the right trick up their sleeve. Somewhere to pool this sort of information and a decent way of navigating it will be very useful indeed.

I also think it has a fairly good chance of working well. I can see possible problems; it would be hard (and probably counter-productive) to set down very firm rules about how articles should be( such as exist at Wikipedia) s conflict resolution will rely on people being reasonable. I think this won’t be too bad though because (IME and compared to academics in other disciplines) mathematicians have, typically, as a group, fairly good habits with respect collaboration.

3. What do you get if you add two ODD numbers together? – an even number
4. Do you think this is always true? – yes
5. Why do you think so? My MoM Said So

The question is what she could possibly be expected to have put that would be better. Could a child so young be taught even a vague understanding of a better reason? Some informal “first draft” of what a proof is? Is it worth the effort to try?

On a side note, sorry for the not posting for a while.

Well I like my job, but I was a bit surprised by this. Then I saw that the methodology is ridiculously arbitrary . Physical exercise is intrinsically bad? I knew there was a reason nobody wanted to be a professional footballer. Apparently meeting the public is also very bad. Yeah, people suck don’t they?

Basically, according to this survey mathematician is the best job available but only because there isn’t full time employment available as the subject in an experiment on sensory deprivation (a job that would actually share many of the downsides of mathematics but would lack the cycles of manic optimism and crushing disappointment).

Oh and $94,160? That’s Zimbabwean dollars, right?

Hat tip to Edge of the American West (who are historians and philosophers amongst other things; pwned!)

My complaint about rephrasing Godel’s result as saying that the Godel sentence G can neither be proved nor disproved is that, to me, the ontological status of “G cannot be proved” is *exactly* the same as the status of “G is true”. To say that G is true is to say that there does not exist a natural number satisfying such and such a property. To say that G cannot be proved is to say that there does not exist a proof satisfying such and such a property. I don’t see how the latter makes any less ontological commitment than the former.

GARGH! Thus (unless someone can furnish me with a reason why it may be wrong) I probably going to have to accept that there isn’t really a good reason not to say “true”. After all there is no alternative rendering that avoids the same potential confusions and we don’t expect popularisations to define things perfectly – just to give the “shape” of the idea, so any confusion between a technical use of “true” and an everyday one is acceptable (they are different but close enough to get some understanding – it’s not like “group”).

It still seems squicky to me but I guess that’s my problem.