What’s a “Mathoscope”?

I’m establishing a few preliminary concepts that are necessary to advance a philosophical system that is likely to be entirely new to you. This is not a trivial system by any means, thus some background concepts are required. Today I’ll tackle “mathoscopes”. This concept is very simple, but I’d like to underscore it right up front. You will see that it has lot of explanatory power about “where we are” and “how we got here”.

Consider the telescope! When we want to have a look at a celestial body, our own eyes are not up the task of fine observation. Thankfully, somebody along the way invented the telescope. Who exactly that was remains a matter of dispute, but nevertheless we now have a device that lets us “zoom in” optically upon distant objects for a magnified view. So, a telescope is a handy observational tool. Now imagine that you are a physicist trying to look into the world of the smallest bits of matter — subatomic particles, waves, fields, etc. These bits of matter defy human perception, they are far too small and tricky for visual observation no matter how much you aid the human eye. The devices physicists use to investigate this world are sets of mathematical frameworks. They are groups of mathematical equations, each coupled with a set of assumptions and sets of rules that constrain how you manipulate the mathematics in each case. Each set of mathematical rules with their attendant assumptions is a mathoscope.

Several groups of mathoscopes are currently operational in modern physics. For example, we have quantum mechanics, which is a name we’ve given to a broad set of mathematical techniques used to describe the action of subatomic particles. Each set of techniques under the category of quantum mechanics is its own mathoscope. An example of one of these mathoscopes is Werner Heisenberg’s matrix mechanics. These are a set of matrix transformations meant to describe the mysterious orbits of electrons, which seem to defy naturalistic concepts of an orbit according to classical orbital mechanics. Many such mathoscopes exist in modern physics, and, worse, whole sets of mathoscopes are completely separate and unrelated to each other. For example, you need one set of mathoscopes for quantum mechanics, you need an entirely different set for gravitation, and then an entirely different set for classical mechanics.   

So the situation resulting from all these sets of mathoscopes is very confusing. Worse, the products that flow from these mathoscopes are very weird. For example, in theories of modern physics, space and time can be thought of as the same entity. Space and time become one, called “spacetime”, because this is what mathoscopes from Relativity Theory suggest. This defies the naturalistic imagination in which space and time are clearly perceived as two completely different things. It’s really not easy to wrap your head around space and time as being aspects of one dimension, yet this is what the mathoscope demands. 

Yet, physicists have perfect faith in the accuracies and abilities of their various mathoscopes. For one thing, it seems that they might have little choice in the matter because they are working in realms with seem to elude perception. You cannot eyeball subatomic particles, for instance. But for another thing, the utilitarian success of Newtonian physics birthed a tradition of faith in empirical laws that conformed to mathematical descriptions. No other field in science has benefited from a more happy correspondence with truth than physics has with its mathematics. Armed with a priori mathematical constructs, physicists found that reality did appear to obey these constructs. The universe, it seemed, really did obey the mathoscope. It’s a short leap from here into confusion if one forgets that the mathoscope is but an observational tool, never a cause. 

But nevertheless physics has always been conducted under the rule that he who has the “best” mathoscope “wins”. In this post, Hans Schantz does a nice job of presenting the Ptolemaic system for predicting planetary motion. This Ptolemaic system is the Ptolemaic “mathoscope” that ruled astronomy for a very long time, from Ptolemy until the Copernicus-Galileo-Kepler trio emerged. As Hans explains, the Ptolemaic mathoscope did the job that it was designed to do very well. That was the real problem for Galileo. He was on to something about heliocentrism, but he wasn’t able to present a better mathoscope to explain it. That would take Kepler, who put forward orbital mechanics, a new, compelling mathoscope. To the mathoscope victor goes the spoils! This is really the utilitarian heart of scientific epistemology in physics. Such is the power of mathoscopes here.

Scientists who use physical lens devices such as telescopes and microscopes are well aware of the framing and the limitations of these devices, say, for example, lens effects. And they account for them. Overwhelming confidence in their mathematics seems to blind physicists with regard to their mathoscopes, however. For an example of this effect, let’s see how we got to this strange conclusion that space and time are one. It all started with a mathematical craze that was raging around the year 1900, for exploring non-Euclidian geometries. What’s a non-Euclidian geometry? Well, let’s take a couple of the fundamental axioms of Euclid and throw them out. One axiom says that the shortest distance between two points is a line. Let’s throw that out. Another Euclidian axiom says that through any point in space, exactly one line can be drawn that is parallel to another line. That’s too wordy! Let’s toss that out, too. If we do these things, we can still construct a geometrical system that works. It is valid, it is descriptive. In fact, throwing out these two axioms is exactly what Bernard Reimann did to construct “Reimann geometry”, which by the way happens to be the geometric mathoscope that best “fits” Einstein’s relativistic mathoscopes. 

In these explorations of non-Euclidian geometries, Hermann Minkowski decided to re-interpret the age-old Pythagorean theorem, the one about the sides of triangle, but with time as a component in addition to the spatial components. Using this new, weird mathoscope, Minkowski looked at the calculations of Hendrik Lorentz about the problem of bodies moving near the speed of light. He found that the regularities described by Lorentz could be expressed in a neat way using this new, weird mathoscope. Einstein ran with it, adopting the Minkowski mathoscope into his mathoscope about gravity. In this way, a transformation of a formula about triangles got folded into additional transformations, and then this transform math got taken as descriptive. Space and time lost their independence and got disguised as one entity. Physicists now believe in “spacetime” because their very weird “winning” mathoscopes tell them so. Lost is the sense that mathoscopes are observational tools; instead, they’ve become tools of divination.

In the midst of all this madness about non-Euclidian geometries, Jean Henri Poincare emerged and he made a stunning observation that really doesn’t get as much attention as it probably should. Poincare offered an article of profound caution, and really a shocking discovery. Poincare observed that an infinite number of valid geometries exist. Hearken back to my post on Zen and the Art of Motorcycle Maintenance where this discovery is discussed in terms of the Infinite Hypothesis Problem. This problem is that if an infinite number of hypotheses exist, the scientific method invalidates itself. You cannot test them all. How are you ever going to know that you’ve got the right mathoscope among an infinite number of mathoscopes? How do the criteria for being “the best mathoscope” ever escape arbitrariness? 

Well, these are profound problems. Any mathoscope can only tell you what it’s designed to discover, and it can only do so following the rules of its own design. It can’t ever really “look” at existence beyond the metrics that it’s tuned for. You can easily get going into the increasingly weird and incomprehensible places that your increasingly weird mathoscopes are taking you, because you’re untehered to reality. You’re tethered to your mathoscopes. That is exactly what is happening in physics. Have a look at string theory, which makes claims about the universe being composed of one-dimensional (yes) “strings”, incomprehensible “manifolds” that require 10 or 12 dimensions, over 10^500 types of spatial vacuums, and so forth. It’s a Wild West of mathoscopes out there!

But, don’t worry. Solutions to all these bewildering problems exist. Or, at least, they can be proposed. I’m going to forward one to you. But, again, it’s neither a brief argument nor is it a trivial argument. Baby steps! It’s as good a place as any to begin by considering mathoscopes. 

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