The idea of mathematics being a sureshot way of describing reality has been postulated and accepted for a very long time. The Greek Mathematician, Pythagoras, gave a convincing argument. Taking an example of a stringed instrument, he explains that when one doubles the length of a string, the note produced by it is an octave lower than the original note produced. More proponents of the mathematical interpretation of reality include imminent philosophers like Aristotle and Plato. The latter did not admit anyone who did not know Geometry into his Academy.

Many arguments validate the above school of thought. For instance, the algorithms that are required to perform operations on our computers is in the heart and soul of physics, wherein the ongoing endeavor to unravel the secrets of the universe requires such high mathematical rigor - that it is convincing to think that “Mathematics is the key to reality”.

However, there are some diametrically opposite things to consider as well. Galileo prophesised that certain things we perceive - such as taste, odour, colours etc wouldn’t exist had the world been mathematical. If one thinks of it, he is right - after all one cannot end up knowing how the color yellow looks by merely learning the wavelength of a particular source of light. Similarly, by knowing which neuron is stimulated to sense taste - one cannot figure out how exactly something tasting sweet feels on our tongue. So, there has to be a middle ground.

One might still wonder, what is it that mathematics brings to the table that elicits radical thoughts like those mentioned above? Does mapping a particular physical process to a mathematical function or any such entity actually mean that the given process can be explained with the help of Mathematics? Let’s find out.

There are four major well-accepted philosophies on what mathematics is. While all of them don’t look at mathematics from the viewpoint of reality, they play an essential role in laying the background for further discussions. The philosophies include:

Proponents such as David Hilbert intended to axiomatise (“formalise”) all of mathematics by creating a large number of “axiomatic systems”,so that it is much easier to solve problems due to the self-contained nature of the rules. Formalism treats mathematics as a set of symbols/numbers, which do not have any actual physical significance. Its results involve manipulating these symbols according to a given set of rules. Each axiomatic system is independent of one another; and each theorem is an independent result. So the notion of applying one branch of mathematics to solve problems in another is quelled right away due to the self-contained nature of axiomatic systems. Also, no links to reality can be established as per this philosophy.

- Logicism is the interpretation of mathematics as an extension of logic,i.e., each mathematical statement could be perceived as a logical statement. These statements could be used to logically prove more things that are mathematically important, while even being able to explain perceivable reality. As per this doctrine, all axioms of mathematics must be interpretable in an intuitive way and be applicable to reality. However, some “uninterpretable” concepts like imaginary numbers are still applicable to the real world. Extending this argument, it becomes hard to interpret more complex mathematical concepts like Abstract Algebra and Non-Euclidean Geometry using simple logic obtained from first principles, but these are still useful in studying the real world.

The theory of Platonism asserts that mathematics is a realm that is independent of the physical world. In other words, the mathematical world exists irrespective of human perception - hence it is non-intuitive in more senses than one. This “abstract” interpretation of what mathematics is, while independent of the existence of reality, doesn’t rule out the latter’s existence. This brings us to the drawbacks of this theory - Plato, who proposed this theory, chooses a conservative approach by not exploring how mathematics could be linked to the realm of physical reality and what we can learn about it. So, something new needs to be brought up which can account for or even perhaps disprove the existence of an explanation of reality with the help of mathematics.

According to this theory, mathematical objects are nothing but mental constructs which form a part of a mathematician’s mind. These “perceptions”, however, are subjective and so is the perception of mathematics. These mathematical constructs, while understandable with the help of analogies that relate to the real world - cannot explain the world as such. Moreover, the interpretation of each of these objects is highly subjective(“It’s all in the mind” - as they say). For example, if one were to study a sphere(a euclidean shape), it could be intuitive for one person to imagine a ball; and for another to perceive it as an orange. Either way, both are studying the same thing.

While these theories explain what mathematics actually is, rather than trying to explain reality, they ignore the existence of reality. This way, the scope of extending mathematics to reality gets diluted. So, a more comprehensive theory needs to come about to answer the function of math and how it is linked to reality.

Instead of looking at mathematics as something that involves a collection of objects, one can look at it as a set of patterns and structures. “Structuralism”, as this theory is called, exemplifies that these structures are made of groups of mathematical objects - with a common thread within them(common thread implies axioms and theorems - rules to bind everything together). For example - a Euclidean Space is a structure where we encounter Euclidean shapes as objects, a set of topological objects is dealt with in topological structures etc.

The structures we talk about here are studied as a whole - with only the relevant interrelations between objects making sense in the study. This implies that structures are very dynamic entities - i.e. objects which mean one thing in a structure can mean something else in another(For example, if one takes an equilateral triangle, it would represent a triangle in a structure of euclidean figures while it would mean something more specific in a set of triangles). This tells us, it is the structure/space that defines the objects present in it. Changing the structure, implies changing the object’s interpretation. Hence, studying the structures instead of the objects in it is something more fundamental, in turn taking us closer to understanding reality.

While these “structures” are abstract, it would be easier to visualize them in terms of sets(after all, the closest one gets to perceive them is set theory). Now, model theory states that for a model to be successful, it has to be true for a given structure - i.e. a model meant for real numbers only holds up for real numbers and nothing else. Objects are just representative examples of what exists in structures.

As far as perceiving reality(worldly objects and phenomena) is concerned, one is able to describe it mathematically because of the underlying mathematical structures present there. Science - which is the study of observable phenomena, uses mathematics to “rationalize” its claims rather than painting a picture of it, so to speak.

“Mathematics doesn’t directly explain reality, it is the language through which reality is explained”.

Why does structuralism work then? This can be understood through a small experiment which can be performed at home. Take a rope and make a figure eight knot following the instructions in the picture. Now, try converting it to an overhand knot( without untying it. Can you do it?

Spoiler alert - No! Topology, a branch of mathematics, has a concept called “knots” which constitute a structure of their own. A simple proof to this end - that these knots are not isotopic shows easily that they can’t be interconverted without tying or untying. One uses axioms and theorems which are relevant to knots to prove the above mathematical fact. Again, mathematics doesn’t demonstrate the actual tying and untying but only rationalizes/ justifies the existence of natural phenomena like above.

Through the presence of mathematical “structure”, one gets a framework to explain physical phenomena more easily. Also, new ways of incorporating structures are sometimes created to explain these phenomena - the very reason mathematical analysis was created. Another advantage of structures is that studying one branch of mathematics can give us insights into the study of another. If the structure is preserved amongst objects, then we can relate their properties too - for instance, the study of real numbers can give us an insight into the study of natural numbers. This point is important for two reasons - one, that the inherent flaw present in the classical approaches of mathematics depicted previously is sorted out. The second reason is when we do science or describe anything in the non-mathematical world, one commonly uses only part of the structure types in question(relevant theorems, axioms, etc.) and not the whole set.

This brings us to another corollary from the example of real and natural numbers we enumerated above - that it is possible to study parts of a structure, which we can call a “substructure”. This concept of a substructure, which scientists apply to their research knowingly or unknowingly on a regular basis, is what gives a perception of the existence of a branch called “applied mathematics”. This branch is often considered different from “pure mathematics” - which looks at these mathematical structures from scratch. However, from the same ideas we have tried to propagate in this paragraph and the previous one, we can understand that applied mathematics is in fact, merely a constrained version of pure mathematics. This fact is important, because this is the very reason a scientist has the license to apply a select few, relevant principles of a given mathematical structure to his research without worrying about the broad, overarching concepts underlying them, thus making his life much easier. This apparent overlap between the two pseudo-branches of mathematics isn’t explained by any of its older philosophies. Thus it creates a solid case for structuralism to be a better philosophical principle to explain what actually is reality as per a mathematician.

Pythagoras is said to have remarked once that “All is number”. While it is not true that the real world is composed of mathematical entities, reality can certainly be explained in a concise, easy to understand way by using the principles of mathematics. The philosophy of Structuralism seems to aid the process, as it provides a proper framework and the required flexibility to apply the existing rules of mathematics or even create new rules(the way Newton did by inventing calculus) in order to study the reality of this world.

https://philosophynow.org/issues/102/Mathematics_and_Reality

https://phys.org/news/2013-09-mathematics-effective-world.html

Shapiro, Stewart. “Mathematics and Reality.” 

Down the Rabbit Hole with a Mathematician: Unveiling the Fabric of Reality

Science often comes across as a highly rational analysis of data. It is not very often that we see the human aspects of science & the implications of any discovery’s use & potential misuse covered in the literature. This is exactly where popular science books shine. One of them being Dr. Siddhartha Mukherjee’s 

The gene is a topic any biology student would have studied. Like fine coffee, the beauty of this topic is that the more you read about it and brew it in your mind, the newer, more profound perspectives you get. A perfect balance of fact and story (that of his family’s history), reminiscent of good ice cream - where a perfect proportion of solid and liquid makes it delectable. Too much ice cream, though, makes you feel giddy, and so does The Gene, as it gets heavy with technical jargon at times.

The story begins with influential thinkers/philosophers like Aristotle and their perspectives on how heredity occurs biologically. Then came Mendel who gave a more definitive theory of inheritance (in the 1880s) through mathematical modelling (a remarkable feat at the time). We eventually learn about the humankind’s venture of playing god by manipulating genes using CRISPR-Cas9 (the very innovation for which Emmanuel Charpantier and Jeniffer Doudna won the Nobel Prize in chemistry in 2020) in the present day. Everything fit neatly into a 500-page book with no rush and no cutting corners whatsoever, in an ever-so-gripping manner - typical of the meticulousness of Mukherjee’s writing.

His writing style is mesmerising- a journey from the simple to the complicated, constant fuel to the fire that feeds the thought process in our brains while reading and not just a melange of words, like a good teacher - one who inspires.

Through The Gene, Mukherjee tackles many sensitive topics like eugenics(i.e selective mating of people in a bid to “improve” the human race) , racism, IQ (the metric), sexual identity etc. and gives the reader a new perspective on them. For instance, Mukherjee calls out proponents of the concept of Eugenics - a gross misinterpretation of how genes work which has wrecked thousands of innocent men and women, not only in Nazi Germany but also in America in the 1910s or 20s, where it led to a national movement wherein “feeble minded” people were castrated - Carrie Buck being one of the prime examples of the same. The hunt for the gay gene, nature vs nurture etc. were other topics ,that were explored with bold, straightforward accounts whilst putting forward an unbiased view.

“The First Derivative of Identity”, in our opinion, is the most profound chapter in the book. Here, he explains twin (through an example of his mother and his aunt who are fraternal twins) studies and that genes do not encode the identity of the person, but the mere “tendency” to turn out in a certain way - twins can be identical - because their genes happen to be more or less identical, yet have very contrasting personalities subject to the environmental factors they face. This simple idea is hands down the most memorable catchphrase in this book. Interestingly though, this is the very chapter which has copped some criticism from the scientific community for overstating the significance of the role of histone modification and DNA methylation in influencing gene expression when in reality, they play a minor role. However, the author clarifies that he does not intend to deny that other factors play a major role in gene expression. He merely emphasized on this aspect to help the reader understand that our genes alone do not define us. This is a crucial message, especially for the average person who does not understand how genes work. It helps the layman understand that, contrary to popular notion, genetic inheritance is not a straightforward process. This sets the stage for the latter part of the book, which deals with the dangers of eugenics.

The Gene talks about how human beings are developing technology at a frantic pace to manipulate genetic machinery to suit their needs - like mass producing insulin using bacteria, curing genetic disorders like Huntington’s disease, schizophrenia (Gene therapy) etc. among others.

The latter forms a part of Human Eugenics, which although seemingly less gory than what we saw in the 20th century around the world, is still marred by safety and ethical concerns. With no proper regulatory framework addressing them, may cause us to push its boundaries too far. Siddhartha Mukherjee warns us that while the gene is a very powerful tool whose manipulation can potentially help the human race live a better life, it is a dangerous thing to have a superficial understanding of how they come together to give rise to a living being.

Genes - as Richard Dawkins puts it, do not contain the blueprint of life, but the “recipe” for life. Manipulating a blueprint modifies parts of something, while manipulating a recipe ruins the whole dish altogether. Introducing a new gene into the human body might bring unnecessary trouble - what if a cancerous mutation shows up? What if it doesn’t give the desired results , i.e; the gene fits in a place where no protein encoding occurs? More often than not, a certain trait is governed by multiple genes - so genetically testing and predicting a potential disease at embryonic stage might not be that easy, or even ethical for that matter.

An interesting example given by the author is that of development of a genetic test for Schizophrenia, a rather complex disease (in origin). Despite being characterised by an inability to exercise self-control and “hearing voices” out of all things, schizophrenics have some good things going for them as well – they are excitable (which is good at times) and have a better drive and desire to get tasks done; as exemplified with an account of Rajesh, the very person who apparently listened to a mind voice which “asked him” to urinate in the open, pulling an all-nighter solving a math riddle while his brothers had given up.

Mukherjee argues that gene therapy must be practiced only when a person is too “abnormal” or “diseased” to even survive. A conundrum arises - is there a real definition of normalcy as far as these things are concerned? In his words, "Normalcy is the antithesis of evolution". Now what is the limit beyond which a person can be deemed exceptionally abnormal? Won’t some sections of the human race end up changing these boundaries as per their convenience? After all, there are two sides to every coin!

These questions are very vital for scientists to ponder upon. A superficial analysis of such techniques isn’t sufficient. Instead, a more in depth & inclusive analysis of its implications is imperative. Overall, this book is an amazing place to start - especially for those who are interested in learning how science has evolved and continues to do so. It could even be termed a modern day classic in some sense.

The Gene - An Intimate review.

I hope this rune[3] won’t be humdrum[4].

The early evolution began with the Big Bang[5],

It started as a Planck – a dense singularity[7].

But with scientific advancements and technical fixtures,

We’re hopefully progressing towards a clearer picture.

They call it the inflation- a burst of expansion.

Deuterium within the first few minutes of formation.

Prolonged cooling paved way for recombination[8],

Followed by darkness, not a child’s fascination.

The effect of gravity gaining prominence,

And after countless millennia, began the sapiens’ dominance.

While we live in an exquisite and ethereal world.

The mysteries that surround us are yet to be unfurled.

The hunt still goes on, will it be CDT[10], E8[11], or the String[12],

That would give wings to the Theory Of Everything[13].

Glossary and References:
1. Incipiency: origination, being brought into
existence.

2. conundrum: an intricate problem/mystery

3. rune : Poem (in finnish)
4. humdrum: boringly monotonous
5. Big Bang: The Universe: Big Bang to Now
6. harangues : here it denotes the public perception(what's thought and widely
accepted)
7. singularity: a location in spacetime where the
density and gravitational field of a
celestial body is predicted to
become infinite. Considered as
the earliest state of the universe during
the Big Bang.
8. recombination: In cosmology, recombination
refers to the epoch at which
charged electrons and protons
first became bound to form
electrically neutral hydrogen
atoms.
9. re-ionization: re-ionization is the process that
caused the matter in the
universe to reionize after the
lapse of the "dark ages.”
10. CDT: Causal dynamical triangulation | Wikipedia
11. E8:An Exceptionally Simple Theory of Everything | Wikipedia
12. String Theory: String Theory | Space
13. Theory of Everything: It is a hypothetical single, all-encompassing, coherent theoretical framework of physics that thoroughly explains and links all physical aspects of the universe together

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Down the Rabbit Hole with a Mathematician: Unveiling the Fabric of Reality

The Gene - An Intimate review.

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