Matematik, ur ett försök till kvalitativt perspektiv

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A DEEPER SIGNIFICANCE
Resolving the Riemann Hypothesis

PETER COLLINS


The key to unlocking the Riemann Hypothesis lies in a qualitative rather than solely quantitative appreciation of mathematical relationships. When viewed in this light it can indeed be resolved whereby it is seen to have the most fundamental implications imaginable for our very understanding of what is meant by Mathematics.
INTRODUCTION

150 years ago in 1859, Darwin's “Origin of Species” was published.

In the same year a brief mathematical paper of breathtaking originality was delivered by Bernhard Riemann relating to the distribution of prime numbers. One observation in that paper mentioned almost as an aside - known since as the Riemann Hypothesis - has assumed such importance that it is now commonly accepted as the outstanding unsolved problem in mathematics.

Throughout its history the Riemann Hypothesis has been the subject of intense investigation by the finest mathematicians. However it has shown itself incredibly resistant to proof with so often, an apparent solution managing to elude final capture in the most tantalising manner. Indeed due to its seemingly impenetrable nature, hints of a more fundamental difficulty can be gleaned through the comments of some of the greatest authorities on the matter.

For example Brian Conrey [1] :

"The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don't understand about the link between addition and multiplication."
And Alain Connes [2] in somewhat similar fashion:

"The Riemann Hypothesis is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication. It's a gaping hole in our understanding..."
Finally Hugh Montgomery [3]:

"Sometimes I think that we essentially have a complete proof of the Riemann Hypothesis except for a gap. The problem is, the gap occurs right at the beginning, and so it's hard to fill that gap because you don't see what's on the other side of it."
It is my firm belief that the failure to solve the Riemann Hypothesis relates to a deep philosophical problem that strikes at the very heart of what conventionally is understood as mathematics.

However to put this in context we need to go back 2,500 years in history to a decisive event that subsequently helped to define the very nature of mathematics in Western understanding.