Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down.

I am wondering shouldn't it be written rigorously as something like: For all $E$ in a given range of values and all $m$ in a given range of values, we have $E =mc^2$? If not, then logically the equation $E = mc^2$ is sometimes valid and sometimes invalid as $E, m$ run through their given ranges, so that one cannot use "$E = mc^2$" to express what it is intended to mean.

Then how does one incorporate this phys-mathematics writing style, as illustrated in the beginning, with the authentic mathematical writing style, as illustrated in the second paragraph, in the sense of logical rigor?

  • $\begingroup$ @Jim Oh thanks. Then would you please help me make the question more physics? Thanks so much. $\endgroup$ – Megadeth Aug 22 '14 at 15:26
  • $\begingroup$ @Jim: Oh I found there is probably a misunderstanding between us! Somehow the first version of my question make you under the impression that my use of "universal" refers to physical universality. Indeed I mean mathematical universality! Please see the edited version!! $\endgroup$ – Megadeth Aug 22 '14 at 15:36
  • $\begingroup$ Shouldn't it really be $E:m\to\mathbb R$? Or $m\mapsto mc^2$? That is, $m$ has a range and $E$ is some value given $m$. $\endgroup$ – Kyle Kanos Aug 22 '14 at 16:36
  • $\begingroup$ Ah Thank you so much. Owing to my shallow understanding of physics, I do not know whether the dependent one or the independent one was intended. But, if there is a function carrying every $m$ to exactly one $mc^2,$ then $E$ is called the functional value of this function at $m$. $\endgroup$ – Megadeth Aug 22 '14 at 17:05
  • $\begingroup$ @Kyle Kanos: And even in functional case, in authentic math. writing one would say either "the function $E = mc^2$" or "For every $m$ there is exactly one $E$ such that $E = mc^2$". Still quantifier problem there. Leaving an equation alone such as writing $z = x+y$ is logically meaningless. For the truth value of "$z=x+y$" is not fixed; as $x, y, z$ varies the truth value of "$z=x+y$" varies. $\endgroup$ – Megadeth Aug 22 '14 at 17:17

Excellent points you make. All true too. The written statement you gave is the way one should understand the rest-energy equation. To reconcile the fact that $E=mc^2$ is not always true when $E$ is the total energy (or rather, rest and kinetic energy), we write the full form of the energy equation, which is generally true: $$E^2=m^2c^4+p^2c^2$$ where $p$ is the momentum of the object.

That way, the logical statement you have written is easily visible. When the object is static, or at rest, its momentum $p=0$ and we recover the famous $E=mc^2$.

In general, this is the same for most physics equations. There is usually an always-true, larger form equation whenever there is a famous sometimes-true equation.

In terms of specifying the range of applicability, we write that in after an equation. Example: $$E=mc^2,\quad 0\le m<20M_{Pl}$$ this would indicate that the equation $E=mc^2$ is only valid for all values of $m$ that are non-negative and are less than 20 Planck masses. Usually, however, values like mass are assumed to have the restriction that $m\ge0$ as usually negative mass doesn't make sense. Additionally, this is just an example; there is no upper limit on the mass value for that equation.

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    $\begingroup$ Back in the 70s we were taught the equation $E = mc^2$ is always true with $m$ being the relativistic mass. Your equation would have been written $E^2=m_0^2c^4+p^2c^2$. $\endgroup$ – John Rennie Aug 22 '14 at 15:38
  • $\begingroup$ Thank you so much, Jim. But it was my fault that I failed to express myself clear enough. Would you mind taking a look at my comment left under the question? $\endgroup$ – Megadeth Aug 22 '14 at 15:39
  • $\begingroup$ @JohnRennie It is rather usual to write $m_0$ without subscript and call it just "mass". Especially in most advanced physics. $\endgroup$ – firtree Aug 22 '14 at 15:41
  • $\begingroup$ Ah, yes. Thank you so much. To be honest, since this question is soft, so intrinsically it cannot be completely answered. Especially for the question "how to incorporate phys-mathematics style with authentic math. style". But the answer is real inspiring. $\endgroup$ – Megadeth Aug 22 '14 at 15:52

Physics literature is, usually, not strict in using mathematical rigour. The overlap of mathematics and physics is mathematical physics, and people in mathematical physics are usually found in mathematics rather than physics departments. If you look at papers published in journals in this field, like e.g. Comm.Math.Phys, J.Math.Phys., Lett. Math.Phys., ... you can expect to find physical arguments treated with mathematical precision, and the statements of proposition and theorems therein are at the level of rigour you may expect from a mathematical text.


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