This short video is a little manic, but generally does a nice job of summarizing the philosophical (specifically, metaphysical and epistemological) discussion as to whether math exists somewhere in the universe (though it can’t be, you know, physically detected) or whether math is just a human construct, a set of ideas, or as the video describes it, a “fiction.”
The video’s narrator describes the “math realist” position as believing that math is discovered, like new species are discovered, “out there” in the world. However, since math’s ideas are not physical and thus can never be directly observed by our senses, then believing in “objective” existence of math ideas requires, well, faith.
The video contrasts math to physics and other sciences, which take as their objects of study things that happen or exist in nature. I’d point out, however, that while we can sense things, like a round-shaped, sticky, sweet-smelling pastry we label “pie,” we cannot sense the measurements of the circumference and diameter of said pie placed in a ratio we call “pi” (also, it’s hard to see how any of the terms just used — circumference, diameter, measurements, ratio, pi, and for that matter, pie — exist except as concepts.
I sometimes tell my students that one way of defining real is by determining whether one would mind being struck by that thing. I don’t want to get hit by a pie, but I don’t see how throwing pi at me could ever hurt.
And so, of course, while physics, chemistry, and biology deal with sensible things, as “ologies,” they are sets of ideas, and so these science concepts, models, theories, and laws may not exist physically any more than math ideas do.
In “Zen and the Art of Motorcycle Maintenance,” Robert Pirsig describes science theories as our contemporary technological culture’s equivalent of earlier cultures’ ghosts — things we think are real but, of course, can’t be seen, etc.
And, of course, none of our words are real, either — depending on how you define ‘real’! If real describes things available to our senses, then words are only interpretations of sounds heard or shapes seen.
So it can get complicated to figure out what’s real when one sees something like IBM’s movie of atoms:
The words “These are atoms. Magnified over 100 million times.” are needed to explain what we are seeing in the video, because what I seem to be seeing is something that looks like little metallic spheres — ball bearings, perhaps. I see ball bearings on a gray background that seems to have shading variations in a gray background.
But we are informed that these are atoms — defined in the press release as “one of the tiniest elements in the universe” — but then, these are not quite atoms. They are “molecules” of carbon monoxide, one atom of carbon and one of oxygen — but this molecule doesn’t look in the movie to have two components.
But “look” and “see” also become problem terms here. The images in the movie aren’t a result of light directly striking a light-sensor (as in a digital camera or in our eyes), but are more like graphs made by information received by the probe in the scanning tunneling microscope. According to the Wikipedia page:
The STM is based on the concept of quantum tunneling. When a conducting tip is brought very near to the surface to be examined, a bias (voltage difference) applied between the two can allow electrons to tunnel through the vacuum between them. The resulting tunneling current is a function of tip position, applied voltage, and the local density of states (LDOS) of the sample. Information is acquired by monitoring the current as the tip’s position scans across the surface, and is usually displayed in image form.
So we’re not seeing atoms. In all likelihood, we can never see atoms. But, OK, let’s call what is detected by the scanning tunneling microscope an atom. But an atom, as we’ve all learned, has constituent parts, which are vastly smaller: protons, neutrons, and electrons. So “atom” just means an organizational level, like, say, a “class” is a bunch of students who gather at the same time in the same room, but there’s no class as a particular physical entity — just particular students.
But of course, any explanations for things we see in the physical world is going to be an idea, an interpretation, which means it’s not the same as the physical world itself. Even these ideas I’m using now are merely ideas, by which I mean they are arbitrary, subject to replacement or revision as we see fit.
In talking about the arbitrariness of ideas with my high school students, some asked why schools teach “fake ideas,” to which I responded, “It’s better to learn fake things than nothing at all.” I’m not sure I would stand by that position all the time, but it’s worth considering from a philosophical and educational perspective.