Naar aanleiding van een lezing van Thomas Hertog aan de KU Leuven deed ik nog wat inspiratie op.
In an effort to elaborate on this analogy between mathematics and the universe from the Platonic viewpoint of mathematicians on their subject of study; this analogy provides a tangible construction of mathematical objects and concepts for which no separate Platonic universe is required.
On the contrary, the universe is perceived as a spatial realisation of abstract mathematical ideas, both created from its most fundamental indivisible building blocks as well as encompassing its most farreaching results and beautiful theories.
From a physicist’s standpoint, a language is required in which to describe the universe, including its most basic structure and its overwhelming complexity and dynamic nature. I believe it would be satisfying for physicists to understand their favourite subject of study in a language encoded in that universe itself.
Of course, when viewing mathematics as a human creation and humans as a creation of the universe, a physicist can argue this last idea is already realized.
However, the majority of the mathematical community would strongly disagree with this demotion of mathematics from an entity on itself to a mere product of evolution of the universe. Having studied mathematics and therefore also counting myself as a member of the mathematical community, I think I’m not too far from the consensus when stating that a mathematician in essence is driven to study mathematics by the profoundness and striking beauty of our beloved subject of mathematics.
What to say about Gödel’s result of incompleteness stating essentially that each candidate axiom system from which to build the whole of mathematics or even merely the theory of natural numbers is doomed to fail to include all possible truths about even a structure as clearly and consisely defined as the natural numbers?
I dare to propose to extend the analogy to a comparison between mathematical statements which are true, but not a logical consequence of an axiom system for the whole of mathematics or even only the theory of natural numbers on one hand and on the other hand the impossibility of current physical theories to provide descriptions of phenomena taking place at the beginning of the universe or better even, at the center of black holes.
It is my guess that it might be the case that both occurrences of the failing of theories, the singularities occurring from both mathematics – true statements, improvable from a chosen axiom system – and physics – spacetime singularities as they occur for example in black holes at the beginning or end of a universe – are manifestations of the same limiting properties of a construction built from basic components, containing enough complexity to create the circumstances in which their predictible consequences fail to clarify the emergence of structures outgrowing the capability to be explained by their creating components.
To me, this feels like emergence in its purest of forms.
As this emergence seems to result from self referencing statements on the mathematical side, namely as in Gödel’s construction of a true statement improvable from a given axiom set; it is my opinion chances are that spacetime singularities can be comprehended as being some form of self reference in the universe.
In a concluding remark, I think it might be useful not to avoid these singularities at the emerging breakdown of theories, but on the contrary to try and incorporate these striking consequences of complexity into future theories of everything. It is my conviction that these concepts of self reference and emergence have a more central role to play, rather than being ignored for their unconstructive and difficult nature.
I am inspired, but also more strongly convinced of the value of the above ideas by a recent lecture held by professor Hertog at Leuven mentioning the incorporation of some form of emergence into String Theory.
Obviously, my relections on Gödel’s Incompleteness Theorem and its relation to the search for a theory of everything are inspired by a lecture from professor Hawking titled “Gödel and the end of physics”. The lecture of professor Hertog also led me to think of these striking similarities of our own expanding universe and the contents of a black hole as seen from within the event horizon.
A larger entity could be thought of to hold some manifold like form containing other universes in the form of black holes itself. The result could be some twisted and bent entanglement of universes featuring for example inclusion relationships among others.
Suppose all these singularities, both mathematical and physical in nature as described above, should be of a self referencing nature, then in essence I think they do not provide any truly independant new theories or views, but merely reflect the power of potential complexity which may emerge from a well defined and clear inference process exposing the consequences of evenly clear and precise building blocks.
If correct, this could be called in my humble opinion a true wonder of nature.
If not inspired, I hope to have been able to amuse you, as I myself am not seldom inspired and amazed not only by the universe itself, but not in the least by those providing the means to try and comprehend this universe we call home. I therefore whish to express my deepest gratitude towards you, professor Hawking and professor Hertog for sharing your knowledge and insights not only with your educated colleagues, but also towards the general public.
De mens is gedoemd om vat op zijn omgeving te proberen krijgen door er structuur en verbanden in te zien, ook als die er in werkelijkheid niet zijn. Bovenstaande hersenspinsels zijn daar een voorbeeld van. de waarde ervan ligt mij eerder in de schoonheid van het idee dan in de eventuele waarheidswaarde of stroken met de werkelijkheid. Anderzijds is elk idee van die vorm, dus het zou zonde zijn om een poging op voorhand af te schrijven. Ik beschrijf hier een analogie die ik zie; die mijn brein heeft geconstrueerd in een poging verbanden te achterhalen tussen verschillende domeinen en ik beweer dan ook helemaal niet dat hier wetenschappelijke waarde aan verbonden is. Wel zou het mooi zijn als dit een kiem is van een idee dat tot enige vooruitgang leidt, op welke manier dan ook en via wie dan ook, in ons begrijpen van een of ander aspect van ons universum.