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Mathematical proof
Infinitesimals
Other mathematical applications
Architecture
A valid mathematical proof is never incomplete, in the sense that it achieves what it sets out to do. In 'A Mathematician's Apology,' G. H. Hardy gives examples of mathematical proofs, chosen from the 'mathematics of the working professional mathematician,' but comprehensible to non-mathematicians - or so he claims.
'I can hardly do better than go back to the Greeks. I will state and prove two of the famous theorems of Greek mathematics. They are 'simple' theorems, simple both in idea and in execution, but there is no doubt at all about their being theorems of the highest class. Each is as fresh and significant as when it was discovered...
'The first is Euclid's proof of the existence of an infinity of prime numbers. [The second example is Pythagoras' proof of the irrationality of the square root of 2, and isn't given here.] The prime numbers or primes are the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...which cannot be resolved into smaller factors. Thus 37 and 317 are prime. The primes are the material out of which all the numbers are built up by multiplication: thus 666 = 2.3.3.37. Every number which is not prime itself is divisible by at least one prime (usually, of course, by several). Whe have to prove that there are infinitely many primes, i.e. that hte series (A) never comes to an end.
'Let us suppose that it does, and that 2, 3, 5,...,P is the complete series (so that P is the largest prime); and let us, on this hypothesis, consider the number Q defined by the formula Q = (2.3.5.....P) + 1.
It is plain that Q is not divisible by and of 2, 3, 5,......P; for it leaves the remainder 1 when divided by any one of these numbers. But, if not itself prime, it is divisible by some prime (which may be Q itself) greater than any of them. This contradicts our hypothesis, that there is no prime greater than P; and therefore this hypothesis is false.' ‚
The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons.'
The symbol ‚ above is what I call the completion indicator. It can be accompanied by a declaration, Dn, to answer the question, 'Completeness in what ways?' If the completion indicator isn't qualified, then the default interpretation is 'fully complete,' by the criteria of the declaration. The proof above is complete in the sense that it establishes, with certainty, the existence of an infinity of prime numbers. A declaration could show that the proof is, though, revisable. G. H. Hardy, in a footnote: 'The proof can be arranged so as to avoid a reductio, and logicians of some schools would prefer that it should be.'
The theme {completion} is also given the symbol ‚ and the context will show the difference between theme and indicator.
The mathematical notion that a continuum can be divided without limit I interpret
as {/division}:- the continuum ~‚. dy/dx as
showing the limit of the ratio 
y/x
as x tends to zero I interpret
again in terms of {completion}. Where the process concerned is the giving
of the smallest non-zero values to x
and y then ‚:- x
dx and similarly for y.
Abraham Robinson's 'nonstandard analysis' translates every statement of analysis
which involves limits into the language of infinitesimals. The theme {completion}
can be used to reinterpret infinitesimals or even translate them.
Other mathematical applications
{continuation} is ~‚. A finite line is an instance of {completion} and an infinite line is an instance of {continuation}.
If a graph has all possible edges between its vertices then ‚.
Factorization of the number 12 ‚ if the factors given are 1, 2, 3, 4, 6 and 12.
Completion in architecture: a comparison of Gothic and renaissance
From Nikolaus Pevsner, 'An Outline of European Architecture,' Page 193 - 195. My emphasis.
'The windows of the Palazzo Rucellai are bipartite as in other palaces, but an architrave separates the main rectangle from the two round heads. The relation of height to width in the rectangular parts of the windows is equal to the relation of height to width in the bays. Thus the position of every detail seems to be determined. No shifting is possible. In this lies, according to Alberti's theoretical writings, the very essence of beauty, which he defines as 'the harmony and concord of all the parts achieved in such a manner that nothing could be added or taken away or altered except for the worse'.
'Such definitions make one feel the contrast of Renaissance and Gothic most sharply. In Gothic architecture the sensation of growth is predominant everywhere. The height of piers is not ruled by the width of bays, nor the depth of a capital, or rather, a cap, by the height of the pier. The addition of chapels or even aisles to parish churches is much less likely to spoil the whole than in a Renaissance building. For in the Gothic style motif follows motif, as branch follows branch up a tree.
'One could not imagine a donor in the fourteenth century decreeing, as Pope Pius II did when rebuilding the cathedral of his native town (renamed Pienza to perpetuate his name), that no one should ever erect sepulchral monuments in the church or found new altars, or have wall-paintings executed, or add chapels, or alter the colour of walls or piers. For a Gothic building is never complete in that sense. It remains a live being influenced in its destiny by the piety of generation after generation. And as its beginning and end are not fixed in time, so they are not in space...'
