As an aside, I’ll throw down a few thoughts that have been rattling around my brain on issues of scientific methodology.
Consider the basic objects of geometry: point, line, circle, plane, cube, sphere, etc. These objects, as Plato said, do not really exist because they are anthropomorphisms, they are idealizations which the human brain is capable of operating on in a consistent manner. Steven Pinker has some lectures available online where he discusses the “intuitive theory of physics” embedded in natural language. The way we use language slices up the physical world into 1-dimensional, 2-dimensional and 3-dimensional idealizations each of which are governed by unwritten semantic rules regarding what can be done to them. Edges and boundaries are reified in human language and spoken of as if they were themselves a thing (idealization of idealizations, in essence).
Using these tools, we can construct new idealizations and this is precisely what mathematics since at least the time of Descartes has done. We can speak of the “tangent” “norm” “inflection point” and so on because these are conventional idealizations that have been built up over the centuries. You aren’t born with these idealizations and you must acquire them through metaphor based on absorption of explanations of them in terms of the idealizations that you were born with. For example, we can explain the “norm” of a surface at a particular point X as “the arrow which is perpendicular to the surface at point X, in every direction around X (all 360 degrees)”. In the case of a sphere, this is roughly visualizable as holding the end of a pencil against a basketball in such a way that the other end of the pencil is “equally distant” from the surface of the basketball in all directions around the pencil.
It is true that the world of mathematics is not the world of physics because the world of mathematics is not constrained by physical law. Infinity is easily conceivable and is extremely useful in mathematical theory. But it is absurd to speak of a physical infinitude. We cannot even imagine what it would mean for a physical quantity to be infinite. It’s basically a contradiction of terms… a quantity is observable because it has boundaries (finite), if it were infinite, it would have no boundaries and could not be observed (and hence, would not be real). To avoid errors, we cannot allow the things that are useful in mathematics to seep into physics just because they are useful in mathematics and don’t cause problems in the idealized world of mathematics.
However, that doesn’t mean that mathematics is useless or ruins physics whenever it comes into contact with it. For example, the world of ordinary experience is, in fact, three-dimensional Euclidean space. We can see this because the theorems of geometry are scale-invariant, at least, up to any size large enough to matter for human beings. Within bounds, it doesn’t matter how big or small you make a triangle, its sides will still conform to Pythagorean theorem. Physical space - within the bounds in which this correlation holds - is actually Euclidean three-space because its behavior is indistinguishable from Euclidean three-space.
But I’m skipping ahead of myself. Clearly, the Universe doesn’t crunch numbers to figure out what to do next. Our mathematical laws describing the physical Universe, however, require tremendous amounts of number-crunching to figure out the state of the system from one moment to the next. This is one of the areas where we’ve mistakenly allowed mathematical idealization to seep into physics. I just said above that physical space is Euclidean three-space but I think this is not true in the sense of the Euclidean three-space of real numbers. I do not believe that physical space is real number space and this disagreement might account for a lot of the absurdities of relativity theory (black holes) and quantum mechanics (point particles).
Rather, physical space consists of ratios, it is rational space. There are so many swings of a pendulum per rotation of the Earth about its axis. This is not a mathematical claim, it is a physical claim. We can count the number of swings of a pendulum per a rotation of the Earth. In so doing, we establish a ratio between the swing of the pendulum and the rotation of the Earth. Time is thus measured as the ratio of two different kinds of events. Space is measured in the same way.
To measure time and space is not to say that “time exists” and “space exists” any more than measuring the surface of a balloon with a string means the “surface of the balloon” exists. All of these things are idealizations and it is the nature of human language to reify boundaries and other intangibles when it is useful to do so. Measuring something does not require us to reify it as long as we’re careful to consistently maintain the fact that measurement is always the act of establishing a ratio between two physical quantities (objects or events). When we say “the ship is 100 meters long” we mean that if you laid out 100 meter sticks end to end, they would be congruent with the bow and stern of the ship.
In order to establish ratios, the magnitudes we are comparing must be countable (i.e. discretely resolvable by a human observer) - we need a whole-number numerator and a whole-number denominator. Hence, scientific measurement always reduces physical quantities to a discrete or digital form but this says nothing about whether the physical Universe actually is discrete or ratio-like (rational).
However, I think we can make a conceptual leap and make a case for a discrete, ratio-like Universe. The same problems that prevent a human observer from resolving the information in the Universe to its “absolute limit” must affect any physical object in the Universe. Why should atoms or electrical fields have access to more perfect information than we can derive with the most careful instruments constructed to extract every last drop of information about the state of a particular variable in the Universe?
There are two ways to answer this. The first is (essentially that given by Quantum mechanics): atoms don’t have better information. A particular atom is unsure about the state information of the rest of the Universe beyond those limits given by the Planck distance, etc. as a result of the uncertainty principle. But there’s a second answer which I like better: atoms can’t have better information. The second answer is better because it kills two birds with one stone… it explains why we see Planck limits and it resolves the problems that arise with real-number space (infinite information density at every point in space).
What do I mean by “atoms can’t have better information”? What I mean is that just like we can only measure the rotation of the Earth down to the nearest whole-number of pendulum swings of our clock, so every physical object is impinged upon by the state information of the rest of the Universe acting upon it down to the nearest whole-number of its sensitivity or “frequency response” to the physical world. Before you think I’ve gone off into la-la land, please understand that I’m very epxlicitly using metaphor here. Clearly, physical objects do not have little pendulums in them by which they are measuring the time elapse of events around them in order to determine what to do next.
Rather, I mean we should think of the physical world in the small a bit like a metaphor of, perhaps, a spinning gear which can be be spun faster by pushing any one of its teeth or slower by pulling on one of its teeth (think of a merry-go-round). The fineness of the teeth on the spinning gear are what determine its sensitivity. The more teeth, the smaller the changes in physical state to which the gear can respond and the faster it can respond to them. The more coarse the teeth, the larger the changes it responds to and the longer it takes to respond to them. The relativistic conception of the world is that the gears have no teeth at all or, conversely, that they have infinitely many teeth. The quantum conception of the world is that the gears have teeth evenly spaced at the Planck distance (again, I’m liberally using and abusing metaphor, here) but then even the quantum physicists still retain gears with infinitely many (or no) teeth because they use point particles, which leads to their own equations breaking down in the limits.
If a gear with very fine teeth were to briefly spin in the vicinity of a gear with very coarse teeth, there could be no response at all when the fine teeth simply do not meet with the coarser teeth because the time involved was simply too small for another of the coarse teeth to rotate into place. Again, I’m speaking metaphorically, I do not believe the physical world is made up of gears. This is what I mean by the idea that an atom (or whatever) can’t be aware of all available state information that is impinging on it from the rest of the Universe. I think this can explain why we observe the (very real phenomenon of) quanta and Plank’s limits and it can banish the use of real-numbers from physical theory which leads to the stealth importation of absurd infinitudes into physics.
To summarize, I would like to make the case that the Universe is inherently ratio-like and the fact that we can only measure ratios of physical quantities is simply a manifestation of this fact about the Universe. This would explain why the Universe is quantized… all physical quantities exist as a ratio which is simply two, discrete, whole numbers. Second, I would like to make the case that the Universe does not at all points “utilize” or “act upon” all the state information which is impinging upon it from all other points of the Universe and that this accounts for why we see Planck limits and could also account for the fact of noise, friction and the 2nd law of thermodynamics (disorder increases over time). State information is simply constantly being lost because it did not propagate due to the insufficient sensitivity of the physical system on which that state information was acting.
Clayton -
