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

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.

EXISTENT PHILOSOPHIES OF MATHEMATICS

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:

  1. Formalism- 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.

  1. Logicism- 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.

  1. Platonism- 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.

  1. Intuitionism - 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.

THE LINK BETWEEN MATHEMATICS AND REALITY

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.

Or in other words, “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.

CONCLUSION

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.

REFERENCES

  1. https://philosophynow.org/issues/102/Mathematics_and_Reality
  2. https://phys.org/news/2013-09-mathematics-effective-world.html

Shapiro, Stewart. “Mathematics and Reality.” Philosophy of Science 50, no. 4 (1983): 523–48. http://www.jstor.org/stable/187555.


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