I dont know if you know the beautiful blue Danube walz composed by Johann Strauß. The spark for the music was provided to him by a dream containing the words ” Viana be glad oh oh why why”. So dreams often contain some hidden dormant breakthrough. Stepping out of the mist into the dawn of a new beauty.

Now my dream : 1+1= 2 or 1.

1+2= 3 or 1 or 2

1+3= 4 or 1 or 2 or 3.

2+2= 4 or 1 or 2 or 3.

2+3= 5 or 1 or 2 or 3 or 4.

5+7= 12 or 1 or 2 or………………..or 11.

It has to do with groups and circles and the fact that any chess piece can be replaced by a musical instrument and any chess game can thus be turned into a musical composition of some sort, and waves combining up to form larger waves, or breaking up into many individual smaller waves.

Thanks for a great website and your time ( youtube lectures etc…)

William.

]]>For your second question, you can perform induction on a statement such as “Either is zero, or there exists exactly one number such that .”

For the third question, strictly speaking is an operation rather than a function, but it can be made into a function once one has a little bit of set theory available: see Example 3.3.2.

]]>In chapter 2 of Analysis I, the object n++ is never defined. Assuming it is a function, it is not stated until lemma 2.2.10 that every positive natural number has a unique pre-image. However, before that, when considering Definition 2.2.1 (definition of addition), the question does come to mind whether a preimage (any, not necessarily unique) can always be obtained from n++. The questions I have are:

1. Does definition of addition (2.1.1) assume that given a number not equal to zero, one can always work out the (unique) “predecessor” to it?

2. In attempting the proof of lemma 2.2.10 (Let a be a positive number. Then there exists exactly one natural number b such that b++ = a), how do I perform induction on a when, by statement of the axiom of induction, the property should be true for 0 but clearly here is true for all a except zero. I have always taken the first element available ( in this case 1) when doing induction. However, here I am trying to work from first principles, hence this question.

3. is “n++” a function? If so, it seems to have been implicitly defined through its characterisation by axioms of natural numbers. Is that a valid way of defining functions?

Many Thanks for your time,

Naveed

]]>Have you ever made a book which consists of a collection of the posts from this blog? If not, is there any way to download all your posts on this blog?

Thanks!

Peter Yang

*[All books published from 2008 onward were initially posts on this blog. -T.]*

i have a general question.is mathematics something that which already exists in nature and you people are discovering it or is it an invented concept….

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