Wednesday, September 28, 2011

New Perspective on Mathematical Proof

I have argued before that correctly speaking - in dynamic experiential terms - that all mathematical proof is subject to the uncertainty principle.
Thus it represents but an especially important form of social consensus that can never be absolute. In fact in certain terms such consensus can prove especially flawed.

Thus for a period in 1993, it was believed that that Fermat's Last theorem had been proved (only for a fatal flaw in reasoning later being discovered). Now the prevailing consensus since 1995 is that this remaining problem has been satisfactorily resolved. So as time goes by - with no further flaws being discovered - we can accept with an ever greater degree of confidence that Fermat's Last Theorem has been proven. However this conviction always remains of a merely probabilistic nature (that is necessarily subject to a degree of uncertainty).

At a deeper level I have challenged the conventional notion of mathematical proof in that it represents just one limited form of interpretation (where the qualitative aspect of understanding is necessarily reduced in quantitative terms).

More formally, in holistic mathematical terms conventional proof corresponds to a linear (1-dimensional) mode of qualitative interpretation (where again meaning is reduced in quantitative terms). However correctly understood we can have potentially an infinite set of mathematical interpretations (corresponding to all other dimensional numbers) where quantitative and qualitative aspects of interpretation - though necessarily related - preserve a distinctive aspect. And in the inevitable relative interaction between both aspects a degree of uncertainty necessarily exists.

So therefore a comprehensive proof requires both qualitative and quantitative aspects (that are necessarily relative). Conventional proof only appears therefore of an absolute nature because the qualitative aspect is entirely neglected.

However recently I have come to realise that there is yet another way in which the uncertainty principle necessarily applies to all mathematical proof.

As we have seen the Riemann Hypothesis represents the important starting axiom whereby the quantitative and qualitative aspects of mathematical understanding can be directly reconciled.

So the Riemann Hypothesis is already necessarily inherent in conventional mathematical aspects. Because of the reduced nature of understanding (where the qualitative aspect of truth is reduced to the quantitative), it is already assumed that if a theorem is proved in general (holistic) terms that it thereby necessarily applies to each individual case (within its class). So for example one we accept that the Pythagorean Theorem is true for the general case (establishing potentially that in any right angled triangle the square on the hypotenuse is equal to the sum of squares on the other two sides) that this necessarily applies in any actual case.

However this implies confusing the potentially infinite nature of the holistic general proposition with the actual finite nature of individual examples. In other words the qualitative aspect of understanding is reduced thereby to the quantitative.

Thus when we properly preserve the unique qualitative distinction of (potential) infinite and (actual) finite notions, we cannot in the absence of a higher authority as it were, automatically infer the truth of the specific from the corresponding truth of the general case.

Now acceptance of the Riemann Hypothesis is necessary to correctly make such an inference. However, as we have seen, because the Riemann Hypothesis supersedes conventional mathematical axioms it cannot be proved (or disproved) from a conventional mathematical perspective.

Therefore the acceptance that a general (qualitative) proof applies in specific (quantitative) terms is correctly based on the validity of the Riemann Hypothesis (which itself cannot be proved or disproved).

That means therefore that mathematical proof is ultimately based on an act of faith (and is thereby subject to uncertainty).

In the highly unlikely case that a non-trivial zero of the Riemann Zeta Function is ever found off the line (with real part = .5) this will pose an interesting dilemma.

We may be tempted to initially maintain that the Riemann Hypothesis has in fact been proven to be untrue.

However this would raise a much deeper problem. For if the Riemann Hypothesis is untrue, then we are no longer entitled to maintain the connection between an actual specific case and the general truth (for the potentially infinite case).

In other words we would no longer be entitled to infer that the demonstrated finding of our errant zero undermines the truth of the general proposition.

In fact it would be even much worse in that we would no longer be able to trust any proposition that has been proved in conventional terms.

So once again the mathematical edifice is ultimately dependent on a supreme act of faith (in the correspondence of infinite with finite notions).
For this reason anyone who believes in Mathematics should hope - and indeed pray - that no errant zero ever crops up for this would rightly undermine faith in the whole enterprise!

1 comment:

  1. I may have resolved the Riemann Hypothesis by looking at numbers as a string.Here's a rough explanation of how it works:

    Riemann's Hypothesis is that all the non-trivial zeros lie on the line ( y = ½ ) and these zeros have something to do with prime numbers. Since Riemann's Hypothesis includes the line ( y = ½ ), let's chose a quantum energy system in which the energy levels are 2. Riemann's equation also included complex numbers. In a quantum energy system Riemann's complex numbers could be represented by zero's ( 0 ) since ( 2 + 0 = 2 ). ( 1 + 1= 2 ) also equals 2. Riemann also went on to say that the prime number locations were influenced by the position of the zeros. To extend our ( 1 + 1 = 2 ) analogy, we now have ( 1 + 0 + 1 = 2 ). We can convert these additions to numbers ( 11, 101, etc. ). We can now say that we have a real axis ( y = 2 ) with infinite numbers with zeros between their ones ( 11, 101, 1001, 10001, etc. ) on an infinite real axis line ( y = 2 ). We can also say that each of the zero's in ( 101, 1001, 10001, etc. ) are on the line ( y = 2) since the numbers ( 101, 1001, 10001, etc. ) are also on this line. Riemann also said that the values of the zeros ( 0's ) on the line are equal to ( ½ ) which is also true when the line ( y = 2 ) is flipped. We could also extend this argument to one which says that the value of the ones ( 1's ) are also ( ½ ) since the reciprocal of one ( 1 ) is ( 1 ) and the fraction ( 1 / 1 ) when added ( 1 / 101 ) has the value ½ on the line ( y = ½ ) . These numbers would also be evenly spaced on the real axis line ( y = 2 ) since each infinite number would total energy level 2 which means the distance between the numbers would be ( 2 + 2 = 4 ) or a spacing of 4. Flipped the distance between the numbers would be one ( 1 ) since ( ½ + ½ = 1 ). Riemann's Hypothesis says that all his formula's non-trivial zero's are on the line ( y = ½ ) and this is what we've proved up to now since the non-trivial zeros are in the numbers ( 101, 1001, 10001, etc. ). If we flip our infinite real axis line ( y = 2 ), we create a real axis line ( y = ½ ) with flipped infinite real numbers ( 1/11, 1/ 101, 1/1001, etc. ) all the way to infinity. If we add ( ½ + 1/101 ) we obtain ( .50990099 ). Prime number ( 11 ) is the 6th prime ( 1, 2, 3, 5, 7, 11 ). ( 11 X .50990099 = 5.6089108911 ) or 6 rounded to one digit. The zeros in these numbers are Riemann Zeros and the 9's are Riemann 9's and can be adjusted to get a more accurate estimate. As a matter of interest prime number 97 is the 26th prime. If you multiply ( 97 X ( .50990099 ) X ( .50990099 )) you get ( 25.22969903 )which is very close to 26. If you adjust the Riemann zeros to form the number ( .509999 ) and do the multiplication ( 97 x (.509999 X .509999) ) you get ( 25.22960106 ) which is further away from 26. You can see from these calculations that if the Riemann zeros are manipulated as Riemann has suggested to obtain the location of the prime ( or calculate the number of primes up to that point ) you can vary the distance to the true position of the prime. The calculation of the prime numbers and their positions are a little bit more complicated than this illustration but this is the basics.