Since all our insights and discoveries in science naturally lend themselves to mathematics, it is interesting to learn that we have no simple guide for correlating mathematical symbols with concepts of ordinary language. What we want to point out here is the inherent limitations of mathematics as a language for physics. For example, our common concepts of reality cannot be applied to the structure of atoms or the elementary particles themselves because they are not real; they form a world of potentialities or possibilities rather than one of things or facts. Therefore, to what extent is a description of atoms at all possible? Can one speak about the atom itself? Physicist Werner Heisenberg remarked in his publication “Physics and Philosophy: A Revolution in Modern Science” (2007) that this is a problem of language as much as of physics, where words and concepts familiar in daily life can lose their meaning in the world of relativity and quantum physics.
The physicist may attempt to explain a mathematical scheme in a language we can understand, and in terms of ideas we are familiar with. The ordinary language is based upon the old concepts of classical physics, and therefore offers an unambiguous means of communication about the set-up and results of measurements. Of course, the expression of a new scientific idea is not just an attempt to find the right words. Sir James Jeans explained in his book “Physics & Philosophy” (1943) that “as science advances, new accessions to knowledge are continuously interwoven into its terminology”. Here a group of new words will be necessitated by a group of new facts about the quantum domain. In addition, a modificiation in the usage of old words is called for by new knowledge of old facts. Physicist Niels Bohr commented on this challenge in his publication “Quantum Physics and the Philosophical Tradition” (1968) where, “we are suspended in language in such a way that we cannot say what is up and what is down.”
Nowhere has this state of affairs been more graphically illustrated than in the development of quantum theory. Even when philosophy uses a word in a precise and unique sense, this sense is often different from that of science. For instance, the new knowledge gleaned by Albert Einstein’s Theory of Relativity prompted us to modify our use of the words “motion”, “velocity”, “simultaneity” and “interval of time”. We also notice that Einstein’s concept of a “signal” is not consistent with the undivided wholeness implied by quantum mechanics. In addition, Bohr argued that words like “position”, “momentum”, “spin”, “space” and “time” refer to classical concepts which also do not fit into quantum theory. Bohr also maintained that, since our language of its very nature is grounded in our day to day commerce with the large scale world, it would be impossible to modify or change it in any significant way.
To a certain extent, a mechanistic worldview persists in our methods of reasoning. The fundamental ideas expressed in our ordinary language of classical physics include the assumptions that time is a line, matter is solid and immutable, objects have a definite position and momentum, and that events happen in space and time independently of whether they are observed or not. Still, in the expansion of scientific knowledge, the language also expands. Concepts such as “energy”, “electricity”, and “entropy” are obvious examples of old terms applied in a broader context. The new quantum term “electromagnetic field” was different from ordinary language, and was not easily accepted by physicists directed to the mechanical motion of matter. Heisenberg remarked that “even in large dimensions, there are many solutions of the quantum-theoretical equations to which no analogous solutions can be found in classical physics.” After all, when Max Planck won the Nobel Prize in physics in 1918 for his quantum theory, the physical quantum of action initiated a new paradigm for relative rather than absolute standards of measurement.
The correlation between the mathematical symbols, the measurements, and the ordinary concepts is by no means trivial, as we cannot fully comprehend the language of quantum mechanics. Heisenberg observed that “if one wishes to speak about the atomic particles themselves, one must either use the mathematical scheme as the only supplement to natural language, or one must combine it with a language that makes use of a modified logic or of no well-defined logic at all”. It is implied that any system using mathematics as its language will be incomplete in this manner. Jeans also agreed that physics cannot embellish the mathematical symbols of quantum activity with their true physical meaning, “for words are unsuited to the expression of accurate or scientific thought, and apparent differences of opinion often result from inadequate definition of the terms employed in the discussion.”