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# Charlene Werner's understanding of math

Who is Charlene Werner, you might ask? Seemingly she is an optometrist who is a great proponent of homeopathy. However, she is mostly well known for her absolutely hilarious pseudoscience, as depicted in this video (don't be surprised if that video doesn't exist by the time you click on it, as Werner is busy sending cease-and-desist demands to everybody who posts the video).

The sheer stupidity of what she says in that video has been totally trashed in several blogs around the internet, so I'm not going to go through everything she says because that has been done already. Instead, I'm just going to concentrate on her misunderstanding of mathematical formulas in physics, and the "E=mc2" formula in particular, just as an intellectual exercise.

The relevant quote is:

You know that when light is energy, right? OK. And he [Einstein] gave us the theory that energy equals mass times the speed of light. E=mc2. OK. If we take that formula, and we think that there's a lot of mass, right? OK. If you collapse all the mass down into the universe, so that there is no space between the mass, do you know how much mass there is in the entire universe? You think you're a lot of mass, right?... Well, the whole universal mass can be consolidated down into the size of a bowling ball. That's all there is in the whole universe. So, how much mass are you? That's right, an infinitesimal amount. So if you take that formula, E=mc2, you can almost cross out mass. So the formula ends up being "energy = the speed of light."

(No, there's a lot more "mass" than that in the universe, but as said, I'll skip commenting on that and the other things she says, and concentrate only on the math.)

Werner seems to have a fundamental misunderstanding of what a physical formula is. She seems to think that it's some kind of metaphor, a vague description of the relationship between two phenomena. She seems to think that "E=mc2" means approximately the same thing as "there's some kind of relation between energy, mass and light". One also gets the feeling that she seems to think that the little 2 (the "square") is just for decoration, put there because it looks fancy, as she just mentions it but ignores it completely.

She seems completely unable to understand that this is not a metaphor or a vague description. This is a mathematical formula. Mathematical formulas are completely rigorous and obey very strict rules. You can't just shuffle things around however you want and expect it to remain valid.

So here's a list of things that she gets utterly wrong:

• Most obviously, you can't simply "cross out" the mass in the formula. She doesn't seem to understand that the "mc2" part of the formula is a multiplication. If the mass is zero (as she postulates), then it would be zero multiplied by c2, which is zero, so we end up with "E=0" (which is correct, if there is no mass). You don't end up with "E=c2" as she suggests.
• Additionally, she just drops the "squared" part in her conclusion, as if it's just some decorative symbol with no significance. Clearly she has no idea what it means (or, more precisely, she might understand what it means when a number is squared, but she has no idea what it means in this formula). She clearly acts as if "E=mc2" was just some kind of metaphor rather than a strict mathematical equation.
• You can't simply drop elements from an equation at a whim and expect it to remain valid. There's one simple rule in physics that can be used for checking this: The units on both sides of the equation must be equal. (This follows directly from mathematics because each element of the equation has an implicit unit factor attached to it, unitless quantities notwithstanding.)

This is one of the reasons you know that, let's say, "velocity = volume" is nonsensical: Because the units don't match. On the left side we have meters/second as a unit, while on the right side we have liters. These just don't match, which means the equation is nonsensical.

The units on both sides of "E=mc2" match, although it might not be immediately apparent. The unit of energy is a joule. On the right hand side we have the unit of mass, kilogram, and the unit of speed, meters/second, squared. In other words, the overall unit of the right hand side of the equation is "kg·(m/s)2".

The unit joule is defined as "N·m" (newton times meter), and a newton is defined as "kg·m/s2". Hence a joule is defined as "kg·(m/s)2", which is the same as the units of the right hand side of the equation.

If you frivolously drop the mass from the equation (in other words the "kg" from the units) the units won't match anymore, which is why you know that "E=c2" (or "E=c" or whatever) is nonsensical.

• You can't just radically change the meaning of one of the elements in an equation and expect it to remain valid. That's what she is doing: Completely redefining the concept of "mass" (to mean something along the lines of "everything that isn't empty space", whatever that might mean in her head) and then assuming the equation is still valid. Again, "E=mc2" isn't a metaphor, it's a strict mathematical formula.
• She is probably confusing "mass" with "matter". What she is really trying to say is "if you take all the matter in the universe and remove all the empty space". However, matter and mass are not the same thing. This is a category error. It's the same type of error as confusing "matter" with "volume". (And no, if you took all the matter in the universe and removed all the empty space, it would be enormously larger than a bowling ball. See neutron stars for an example of exactly this.)
• Not surprisingly, she doesn't understand what the formula is saying. It's a direct relationship between mass and energy. Roughly speaking, the formula is saying "if you have this much mass, you have this much energy". Or in other words, "if you wanted to have this much mass, you would have to put this much energy in it". At a more fundamental level, the formula is saying that mass is energy (and what the relationship is between the amount of mass, in kilograms, and the amount of energy that is, in joules). By saying "there's practically no mass in the universe so we can cross the mass out" she shows that she has no idea what the equation means at all.
• An even more obvious sign that she doesn't understand the formula at all, a misunderstanding at a very fundamental level, is that she confuses "the speed of light" with light itself (as she proceeds to talk about the importance of light and the human visual system, as if it was somehow related).

The "speed of light", c, is just that: A certain speed. It has nothing to do with light itself (other than that light travels at that speed in vacuum). It's called "speed of light" because light (or, more precisely, electromagnetic radiation) is the most obvious phenomenon that travels at that maximum physical speed. (Any massless particle would travel at that same speed too, as a direct consequence of the theory of relativity.) This quantity could be called something else completely (such as "universal maximum speed" or whatever).

Even if the quantity c would be inherently tied to light and nothing else, equating them would still be utterly wrong. The speed at which something travels is not the same thing as that something. It's like saying that "the speed of that car" is the same thing as "that car". Obviously this is pure nonsense. It's a fundamental category error.

Again, it clearly seems like she thinks of "E=mc2" as some kind of vague metaphor, like a cryptic utterance of a wise prophet which has to be interpreted. No, it's math, not a metaphor.

If there's any deeper meaning to the E=mc2 formula, it's that it establishes an equivalence between energy and mass (rather than energy and light, as she postulates). It gives the exact ratio between the two (in other words, "this many kilograms of mass are equivalent to this many joules of energy".) And this is an exact, mathematical equivalence, not a vague metaphor.