I had a letter printed in the September 1997 issue of Physics Today. In that same issue, they printed a letter by Lorenzo de la Torre about the relation between physics and mathematics. This is a copy of e-mail I sent to him in response. I think it stands on its own.
I read your letter in Physics Today. In physics we try to think up explanations for what we observe. Mathematics is just a tool we use to do this. In order to be useful, it has to be self-consistent, so when we think up something new in mathematics, it has to not contradict anything previous. Therefore the mathematics of the 4th Century B.C. is as true today as it was then, wheras a physics paper that is five years old would be useless to anyone other than a historian. When you say that Euclidean mathematics is a limiting case of non-Euclidean mathematics, in which the curvature is zero, you're saying it's still true for that case. When you say Newtonian mechanics are a limiting case in which relavistic effects are neglible, you're not saying that it's true, you're saying the inaccuracy is very small.
When you say that fundamental particles, black holes, etc. are not directly perceived by our senses, that's true for almost everything. New York City has probably never been perceived by your senses, although you've seen pictures of it, and seen it on television. We detect a tiny amount of what we can safely assume exists with our eyes, ears, noses, and senses of touch and taste. We can make machines that can make what is normally detectable by one sense detectable by another sense. This is like watching sound waves on an oscilloscope. Similarly, we can make machines that can make detectable to one or more of our biological senses, things that can't be detected by any of our senses. This is like listening to the clicks of a Geiger counter. At the outer edge of what we can detect with machines, you have particle accelerators, gravity wave detectors, the Hubble Telescope, etc. We try to think up explanations that could possibly explain what we observe, however we observe it. If there was something we didn't observe, we wouldn't think up an explanation for it. Therefore all of the explanations we have in physics must at least partially explain some aspect of what we observe. Otherwise we would not have thought it up in the first place.
We thought up the neutrino to explain what we observed. That's all physics is. We try to think up explanations that could possibly explain what we observe. You might say these explanations aren't really true. Well, of course not. They're the product of our brains. In the late 19th Century, people thought up atoms to explain the compressibility of gases. You might say that the theory turned out to be true. However, people at that time imagined atoms as tiny spheres, like tiny marbles, which is obviously not true. Therefore, their view of the Universe wasn't true. Similarly, our view of the Universe isn't true either. It would be rediculously arrogant to imply otherwise. In the future, people will view neutrinos and quarks very differently than we do today.
The data coming from the particle detectors of the most advanced accelerators is what it is. Something causes it to be what it is. Who knows what that something may be? It's probably something we could never imagine. However, we try to think up something that could be a possible explanation for that data. We try to think up something that could possibly explain what we observe, and doesn't contradict what we observe. We can never actually achieve that but that's what we strive for. All explanations in physics, no matter how esoteric they may sound, are ultimately motivated by observation. If you theorized a completely undetectable particle, that explanation would not contradict anything that you observe, but it would not explain anything that you observe, so there would be no benefit to the explanation. The editor who wrote, "Yes, Virginia, there is a Santa Claus" said, "Just because you can't see something doesn't mean it's not there." However, if you can't see something, there's no reason to say that it might be there.
When Mangano and Trippe said, "The existence of the sixth quark has become an absolute theorectical necessity", they meant that it was necessary in order to maintain the simplicity of the theory, which of course they would rather. You could come up with a model without a sixth quark but the mathematics would be vastly more complicated. It wasn't just aesthetic considerations that motivated a desire for a sixth quark. The top quark explains bottom quark decays. If b quark were a singlet, it would have no W interactions. It would be possible, but very difficult to explain this without a top quark, so we'd rather not.
Mathematics describes relationships between quantities, whether it's F=MA, or dozens of pages of complex integrals, and therefore is an invaluable tool to physics. However, mathematics doesn't enable you to think up whatever it is in the first place. Physicists casually invent new mathematics as the need arises for it, such as the delta function or whatever. If you see a pattern, such as in the properties of particles, you can take advantage of the pattern that you observe, and hope that it's continued into the realm that you can't observe. You use mathematics to describe the pattern, Lie algebra or whatever, but that does not mean that mathematics is guiding your theory, or is motivating physics. Just because there's a pattern, doesn't mean that it's contonued throughout. The up and down quarks are much closer in mass than what would be predicted by the pattern that exists among all of the other quarks. Mathematics exists only in the human brain, computer memories, or marks on paper. Physics is motivated by observation, meaning something that's actually out there in the Universe.