At first glance, mathematics is quite nice: a definite, reliable, and self consistent field of study independent of time itself. It has this nice logical foundation and proof based approach to truths.

“Behold these symbols so bright,
With definitions laid out in sight,
Follow these rules we have set,
Secrets impossible to forget,
Knowledge beyond this existence,
Is where all mathematicians met.”

~ What my maths tution sir never told me

At second glance, there is a choice at the heart of mathematics that often goes unnoticed: the choice of axioms, or basic assumptions, is both optional and until now, was necessary.

Zero is a number, this we know, A foundation for all to grow.
Successor for each, we do find, No two with the same, of this kind.
Zero is not a successor, clear, In a set with all, it does appear.

~ Chat GPT reciting Peano’s axioms

At third glance, the cardinality of that choice set is atleast as large as the cardinality of space time itself.

There are even bigger problems within mathematics and its relationship with reality. The vast majority of modern physics is now based on mathematical models. But when neither light, nor atoms, can escape a black box in reality, how do we take a look?

How can we even begin to understand it with mathematics? How can we forecast it with maths? Somewhat similar to card counting or crypto: it works when the deck and odds are in your favor. In many ways, this is similar to the concept of probability in gambling. We can use mathematical models to forecast outcomes, but we still don’t know for sure what will happen in a particular one.

So these black holes in the maths of physics and physics of maths, are they for real? Or are they simply kinks in mathematical space time: a limitation of maths itself and not necessarily a limitation of physical reality? It rhymes with a similar classical philosophical question, “If a tree falls in a forest and no one is around to hear it, does it make a sound?”.

How can we be more sure?

In the early 1900s, in fact in that very year 1900, great mathematician Hilbert listed out the most important unsolved problems in mathematics. What he emphasised further was a need to organize mathematical reasoning. He claimed that a formal axiomatic system was needed, which should be both `consistent’: it is free of contradictions, and `complete’: it at least represents all the truths that are known to be true. Hilbert further argued that a mathematical problem should be “decidable” in the sense that there exists an exact procedure or set of instructions however complex to decide whether a proposition is True or False … an algorithm.

Plenty of Hilbert’s problems are now solved.

Over the course of February 2023, I will write 27 blog posts with stories and theories, thoughts and feelings, history and mystery, covering various mathematical singularities. Each theme is about a place in maths where things start to break down. Much like black holes, we do not understand these places well yet. Likely that one of these places is where maths itself ends and reality begins.

Tomorrow we will visit an abandoned hotel and visit the first character of our endless 27 part story about maths.

You can hear the laughter, you can hear the math,
In Hilbert’s Hotel, there’s no escape path,
For every room is filled, it’s always true,
And you’ll always find a place, that’s up to you.

~ Chat GPT’s Tripadvisor review on Hilbert’s hotel


One response to “Singularities”

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