---
title: "Relationship between mathematics and physics"
chunk: 1/2
source: "https://en.wikipedia.org/wiki/Relationship_between_mathematics_and_physics"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:40:16.838189+00:00"
instance: "kb-cron"
---

The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since antiquity, and more recently also by historians and educators. Generally considered a relationship of great intimacy, mathematics has been described as "an essential tool for physics" and physics has been described as "a rich source of inspiration and insight in mathematics".
Some of the oldest and most discussed themes are about the main differences between the two subjects, their mutual influence, the role of mathematical rigor in physics, and the problem of explaining the effectiveness of mathematics in physics.
In his work Physics, one of the topics treated by Aristotle is about how the study carried out by mathematicians differs from that carried out by physicists. Considerations about mathematics being the language of nature can be found in the ideas of the Pythagoreans: the convictions that "Numbers rule the world" and "All is number", and two millennia later were also expressed by Galileo Galilei: "The book of nature is written in the language of mathematics".

== Historical interplay ==
Before giving a mathematical proof for the formula for the volume of a sphere, Archimedes used physical reasoning to discover the solution (imagining the balancing of bodies on a scale). Aristotle classified physics and mathematics as theoretical sciences, in contrast to practical sciences (like ethics or politics) and to productive sciences (like medicine or botany).
From the seventeenth century, many of the most important advances in mathematics appeared motivated by the study of physics, and this continued in the following centuries (although in the nineteenth century mathematics started to become increasingly independent from physics). The creation and development of calculus were strongly linked to the needs of physics: There was a need for a new mathematical language to deal with the new dynamics that had arisen from the work of scholars such as Galileo Galilei and Isaac Newton. The concept of derivative was needed, Newton did not have the modern concept of limits, and instead employed infinitesimals, which lacked a rigorous foundation at that time. During this period there was little distinction between physics and mathematics; as an example, Newton regarded geometry as a branch of mechanics. 
Non-Euclidean geometry, as formulated by Carl Friedrich Gauss, János Bolyai, Nikolai Lobachevsky, and Bernhard Riemann, freed physics from the limitation of a single Euclidean geometry. A version of non-Euclidean geometry, called Riemannian geometry, enabled Albert Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity.
In the 19th century Auguste Comte in his hierarchy of the sciences, placed physics and astronomy as less general and more complex than mathematics, as both depend on it. In 1900, David Hilbert in his 23 problems for the advancement of mathematical science, considered the axiomatization of physics as his sixth problem. The problem remains open.
In 1930, Paul Dirac invented the Dirac delta function which produced a single value when used in an integral. 
The mathematical rigor of this function was in doubt until the mathematician Laurent Schwartz developed on the theory of distributions.
Connections between the two fields sometimes only require identifying similar concepts by different names, as shown in the 1975 Wu–Yang dictionary, that related concepts of gauge theory with differential geometry.

== Physics is not mathematics ==

Despite the close relationship between math and physics, they are not synonyms. In mathematics objects can be defined exactly and logically related, but the object need have no relationship to experimental measurements. In physics, definitions are abstractions or idealizations, approximations adequate when compared to the natural world. In 1960, Georg Rasch noted that no models are ever true, not even Newton's laws, emphasizing that models should not be evaluated based on truth but on their applicability for a given purpose. For example, Newton built a physical model around definitions like his second law of motion 
  
    
      
        
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 based on observations, leading to the development of calculus and highly accurate planetary mechanics, but later this definition was superseded by improved models of mechanics. Mathematics deals with entities whose properties can be known with certainty. According to David Hume, only statements that deal solely with ideas themselves—such as those encountered in mathematics—can be demonstrated to be true with certainty, while any conclusions pertaining to experiences of the real world can only be achieved via "probable reasoning". This leads to a situation that was put by Albert Einstein as "No number of experiments can prove me right; a single experiment can prove me wrong." The ultimate goal in research in pure mathematics are rigorous proofs, while in physics heuristic arguments may sometimes suffice in leading-edge research. In short, the methods and goals of physicists and mathematicians are different. Nonetheless, according to Roland Omnès, the axioms of mathematics are not mere conventions, but have physical origins.

=== Mathematics is physics ===
A well-known dictum of the Russian and Soviet mathematician Vladimir Arnold is "Mathematics is the part of physics where experiments are cheap". While the  phrase generated controversy and even parodies, Arnold has defended it.
Mathematicians Arthur Jaffe and Frank Quinn have noted trends in mathematics towards more focus on intuition even at the cost of rigor and suggest this trend is due to interactions between math and physics. In the framework of Willard van Orman Quine's epistemological holism, our beliefs, even in mathematics, are subjected to the "tribunal of experience", just like in physics.

== Role of rigor in physics ==