Continued from:

6. A mathematical overview

7. Back to biology… via physics

If you skipped the previous section, it can be summarised with the following points:

• It is possible to construct a programming language that is ‘smooth’, like sine waves.
• It is also possible to make this programming language multi-dimensional; a natural pairing with ‘vector spaces’. If you’re a programmer, imagine a multi-dimenional array, or table, in which code is stored in each cell. Now imagine this array can add pieces of code together, and even multiply them, according to a law we define.
• It may be possible to build types of multi-dimensional array that form Lie algebras and Lie groups. These are mathematical constructs often used by physicists, and explained in more detail below.

So, what does this mean for our understanding of nature?

We know that nature is concerned with some of the same problems computer scientists and physicists are; information and symmetry in particular.

Life can be understood through the propagation of this ‘biological information’. The spread of information against the inevitable forces of disorder, or ‘entropy’ is one way we can define what life is. Physicists such as Erwin Schroedinger reflected on this in the 20th century, most notably with ‘What is life?’. It was written long before DNA was discovered, and inspired Watson and Crick to search for it.

Modern physics is deeply concerned with the mathematics of symmetry through ‘Group Theory’. In group theory, we look at how elements of a set are related to one another, perhaps through transformations (this is what Mathematicians term ‘symmetry’). For example, we can study the different configurations of a Rubiks Cube to determine the maximum number of turns required to solve one.

This same branch of mathematics gives rise to our understanding of particle physics, and the Standard Model. The Standard Model is an extremely successful theory; it has given rise to a mountain of experimental evidence and validated predictions.

In contrast, our understanding of biology is nowhere near as robust. The only framework we have to understand biology is evolution, which so far has not yielded mathematical insights, in the style of physical theories which contain mathematical frameworks that generate experimental predictions.

In physics, group theory has proven a valuable analytical tool in its ability to show how particles are related to one another.

What if it could be employed to understand how biological organisms are related to one another?

What if we can analyse evolutionary processes through a more mathematical framework?

I’ve provided some conjectures below - some will only be intelligible to those who have understood qualitatively the previous section. Others can be understood by any scientifically minded reader.

Conjectures

• A geometric framework was conjectured by Andras J. Pellionisz to unify neuroscience and genomics, under a mathematical formalism. I further conjecture that such a framework can be built on top of an algebraic lambda calculus descended from Thomas Erhard’s differential calculus (henceforth, known as Λ)

• Within this framework, organisms can be seen as moving through a multi-dimensional fitness landscape. Evolution’s algorithm moves organisms through this fitness landscape, in a similar fashion to the optimisation algorithms of machine learning.
  • To most biologists, this is a fact. What I conjecture is that this fitness landscape can be described by Λ , and that some of this fitness landscape is encoded within the genome of organisms.
  • It can be shown that within this fitness landscape, speciation will allow life to escape local minima, to find a global maxima.
• If genes are ‘functions’, these functions are not currently intelligible to human beings in the same way that machine learning functions are not intelligible by human beings. We can of course reverse engineer qualitative models to describe them (the gene regulatory networks we know of), but these models are crude in computational terms.
  • Λ may potentialy provide insights to build more concrete computational descriptions of gene regulatory networks, and of machine learning models.
    • We may be able to produce human intelligible, semantic ‘programming language’ from Λ to describe both these phenomena
• Within Λ, genes can be seen as members of Lie groups and Lie algebras. We can thus analyse genes (who have a ‘desire’ to preserve their informational symmetry) through a similar process to which we study particles in modern physics.

For now, that’s it. This blog is a work in progress and I hope to update it with more details on existing content, but also my research as it emerges into a concrete formalism.

-Yaseer