Mathematics Vs. Physics: What Makes Them So Different? - A whiteboard-style graphics illustration displaying a placard that reads "Math Vs. Physics".

I was recently listening to an old lecture from renowned physicist and genius, Richard Feynman. His philosophical musings about how the mathematician differs from the physicist and vice versa set me thinking.

Using Feynman’s brilliant analysis as the starting point, I started wrapping my thoughts around this topic and wrote them down.

This eventually led to this very essay you are reading, where I discuss the fundamental differences between the two fields, the origin of conflicts, and where the boundaries lie.

Towards the end, I also discuss possibilities of how we could bridge the gap in understanding between the two fields, and make the case for strength in divergence.

This essay is supported by Generatebg

Mathematics Vs. Physics: An Interplay of Abstract and Concrete

To begin, we must acknowledge the fact that for science to progress, the two fields of mathematics and physics need to necessarily intertwine as frequently as possible. Their interplay, however, is more complex than it may appear at first glance.

As Feynman sharply observed, mathematics is the language of the abstract, and thrives in the realm of axioms and structures of reasoning. Mathematicians dwell in a world that does not force them to assign concrete meanings to their symbols and phrases.

The beauty of mathematics lies in its abstraction, where, for instance, a theorem is equally applicable to a three-dimensional space as it would be to an n-dimensional one (generality).

The mathematician, hence, has no need to concern himself with the “real world”. Rather, he thrives in an abstract universe, the underpinning of his reasoning not reliant on the application, but the structure itself.

Physics, on the other hand, represents our scientific understanding of the physical world around us. It lends meaning to every symbol, every term, and every phrase. It might draw from the abstract toolbox of mathematics, but it does so with the objective of explaining concrete phenomena.

Physicists often concern themselves with special cases, like the application of a mathematical theorem to three-dimensions. Furthermore, they might even tweak each variable in the said theorem to precisely represent a “meaningful” phenomenon we observe in the real world.

The Case for Special Cases: A Physicist’s Playground

So far, I have established that mathematicians tend to crave generality, whereas physicists tend to immerse themselves in special cases. The physicist’s query is firmly tethered to the reality that surrounds us, whether it pertains to the forces that govern our world, the concept of mass, or the principle of inertia.

To paraphrase Feynman on this point, Physicists thrive on the “seat-of-the-pants” feeling about the world, with each observation serving as a guide to unravelling the intricacies of the universe.

In the realm of physics, symbols (variables) do not exist in a vacuum — they connect to the tangible, the real, and the observed. The complexity of a physicist’s equations emerges not from the pursuit of mathematical rigor but from the attempt to accurately model the physical world around us.

Mathematics Vs. Physics: A Notable Source of Conflict and Issues

In today’s modern scientific playground, there exists ample freedom of development for both fields (more on this later). But still, I often observe conflicts and issues arising from a subtle but notable pattern.

When a clever physicist starts learning mathematics or a clever mathematician starts learning physics, conflicting views and arguments start arising between these fields.

Mathematics Vs. Physics: What Makes Them So Different? — A whiteboard-style graphics illustration displaying a placard that reads “Math Vs. Physics”.
Math vs. Physics — Illustration created by the author

Don’t get me wrong. Gaining knowledge in another field is a great way to gain more context around the reasoning of your cross-field colleagues. But at the same time, clever people often fall into the trap of thinking they know more than what they actually know.

I have covered part of this phenomenon extensively in an essay titled: “Why Do Stupid People Think They Are Smart?” I have also covered potential solutions to this issue in that essay. Check it out if you are interested.

Mathematics Vs. Physics: Bridging the Gap

One of the more recent developments in the field of mathematics is the sub-field: applied mathematics. This field strives to connect mathematics to the real world.

At this point, I must admit that I personally have a passion for this sub-field of mathematics as it has its roots in “reality” and is not “out there” like, say, number theory or modern geometry.

Furthermore, applied mathematics does not just bridge mathematics to physics, but also a vast array of other fields that are relevant to our real-world experience.

While this sub-field is a great boon to both fields, it is easy for us to fall into the trap of thinking “bridging the gap” is the only way to go. I personally do not think so. I see strong value in divergence as well.

Independence in Interdependence: Setting Boundaries

Mathematics and physics, despite their interplay, tread their paths independently. Mathematicians are free to explore their abstract spaces without the responsibility to serve the needs of physicists.

They are not slaves to physics but partners in the act of exploration, offering their axioms and theorems as tools, not instructions.

On the other side, if physicists need a special case, or a simpler, more tangible theorem, they bear the responsibility of deriving it themselves. They cannot demand mathematics to bend to their needs. The mathematician offers the skeleton; the physicist must add the flesh.

Conclusion: Divergence is Strength

To summarise what I have discussed so far, mathematics relishes in its realm of abstraction, creating a universe of reason and logic that can be as general or as specific as desired.

Physics, conversely, deals with the tangible world, deciphering the language of the universe as it manifests in the physical world around us.

Together, these two disciplines weave the tapestry of our understanding of reality. They are distinct, yet intertwined in a partnership that has continued to unlock secrets of the universe, from the microcosmic particles to the expanses of cosmos.

The interplay between mathematics and physics is a testament to the diverse paths of human inquiry. As we continue to delve deeper into the mysteries of the universe, I strongly feel that it is in their divergence that mathematics and physics truly illuminate our path towards understanding the world.

For science as a whole to progress, each field must be pushing its boundaries without regard to other fields. This might be a bit of a controversial statement, but I believe that this is the right way forward.

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Further reading that might interest you: 

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