Abstract Realism

Learning to count is every child’s first introduction to mathematics. In parallel with learning the order of the words that denote number, the association of number with actual physical objects constitutes the learning target. Put down one block, then put down another block and you have two blocks. Another makes three. Take away two of the blocks and you have one remaining. Thus addition and subtraction are absorbed in a very concrete way.

Take away the last block and you have zero blocks. But wait a minute! What is the concrete representation of zero blocks? You can “see” zero blocks – but how do you know it’s not really zero elephants? Or zero watermelons? You have just entered a new zone of perception – the abstract. You know it’s zero blocks because you have asked yourself the question “How many blocks do I see?” The meaning of the actual reality before you is modified by a purely mental process, the (perhaps tacit) posing of a question which sets a context for the answer. There is no concrete representation of zero blocks without this additional mental step.

Concrete, as in the opposite of abstract, is an interesting word. Abstract is a good abstract latinate word, but concrete, well it’s also an abstract latinate word. The Romans made really good concrete – there are Roman roads and aqueducts that have lasted two millennia. Modern concrete is not expected to last one tenth as long, although it has other properties that make it ostensibly useful. Look at this picture and try to imagine what the ruins of Manhattan will look like in two hundred years.

Anyway, having stepped off the shore of concrete representation into the ocean of abstraction, the next step in your indoctrination into mathematical thinking is negative numbers. You might picture a field with zero cows in it by imagining a field with one cow in it, and then removing the cow. But there’s really no way to make a concrete picture of minus one cow. If you are destined to be any good at all at math, you will use a picture in your mind of a number line. There is a spot in the middle of the line labeled zero, and the negative numbers increase from there in one direction (probably to your left), and positive numbers in the other direction on the line – off into the distance as far as you want to imagine. All of mathematics is then available to you, limited only by your motivation and ability to stay faithful to the game.

This intersection of abstraction and reality is the point of power of the human race. It is what has enabled us to dominate every ecological niche and reach out off the planet in a blink of geological time.

Philip K. Dick said that reality is that which, when you don’t believe in it, is still there. There is a reality “out there” – if you don’t believe that then you have bigger problems than that you are spending your time reading this blog. But our ability to make sense of our senses via abstract thought – our reality “in here” – is what makes humanity, for better or worse, so powerful.

I’ve been reading a marvelous little book called “The Marriage of Sense and Thought” by Stephen Edelglass, Georg Maier, Hans Gebert and John Davy. In fact I read it once, and then just a few weeks later read it from cover to cover again, because I couldn’t stop thinking about it. As you might expect from a book with such an overabundance of authors it’s a little uneven, but it’s so worth plowing though the somewhat heavy-handed pedantic parts to get the gems of insight within. It’s a thoughtful realization that “reality” encompasses a number of quite different things (almost none of which are things), all of them more or less, um, real.

It was fascinating to read the details of how it was the work of Galileo (not Descartes for example) that was central to modern scientific thought. He uniquely integrated in himself two kinds of expertise that had been separate non-communicating disciplines since antiquity, the craftsman and the scholar, what we now think of as the experimental and the theoretical. Moreover, his disagreement with the church was even more fundamental than the myth of Galileo we all learn.

I have to quote from the book here:

Galileo broke radically with the Aristotelian tradition when he imagined a pendulum swinging in a vacuum without friction, following exactly the mathematical laws used for constructing its path. Moreover, he insisted that the mathematical laws discovered by Copernicus were actual realities. Copernicus … did not publicly claim any more reality for his circles and epicycles, with the Sun at the center, than for those of Ptolemy, with the Earth at their center. But Galileo insisted that the Sun was at the center in reality. This claim upset the whole structure of the hierarchies surrounding the Earth and implicitly undermined the authority of the church, which was modeled on that structure.

And again:

The conceptual and mathematical skills at the disposal of the Greeks were entirely adequate for them to have anticipated Galileo and Johannes Kepler by two thousand years or more. But they were not interested in practical applications. … Their attention was directed toward spiritual worlds, since the material world was considered not worthy for free men to work in.

But the real heart of the book concerns our bodily senses, and the ways in which we interpret them to conceive of and embrace the reality we find ourselves embedded in:

In our naïve everyday experience we assume that the world as a coherent totality – including objects and their spatial relationships – is simply given to us through sense perception. We are quite unaware of our inner participation in apprehending the world. This commonly held, naïve assumption is the starting point for the considerations that ultimately lead to the objectivist worldview.

I was gobsmacked but persuaded by the argument that electromagnetic fields, for example, are only real in the sense that negative numbers are real, as an abstraction. We are taught to think of them as little tiny balls of stuff, photons – but there is no stuff – or else waves, like water waves but there’s no equivalent of the water for the waves to be waves of. There’s no such thing as electromagnetic fields, in the same sense that there’s no such thing as negative-two cows. And there is, in the same sense that there is.

This works for physics, sort of, if you’re willing to abstract away time and consciousness. But it’s a serious failure in biology, where the insistence on ignoring what we can plainly see though not measure quantitatively causes a self-induced blindness, as Steve Talbot has been so wonderfully expounding on.

I can’t really capture anything but a hint of the clarity and richness of the presentation of ideas in this book. You should read it.

Written on June 2nd , 2021 by Victor Grey