I mostly think about abstract ideas. I study machine learning and natural language processing, so I spend all day learning calculus and linear algebra (abstract) to try to teach computers what the word “happy” means (also abstract). That way, when you ask Siri or Alexa to find you a happy family photo, they’ll actually be able to do it.

Most of my problems come from the fact that these things are abstract, and therefore hard to define. For example, to teach a computer what “happy” means, first you have to be able to define “happy” in terms of numbers and pixels. This is non-trivial. Some might say it’s impossible; “happy” is too abstract to really define concretely. It’s an idea that lives in the “abstract plane.” (And yet you could probably find me a happy photo if you tried.)

For a large class of cases … it can be explained thus: the meaning of a word is its use in the language.
- Ludwig Wittgenstein

Similarly, mathematical abstractions caused me a lot of headaches. I never seemed to be able to ground myself when talking about mathematics; it felt like math also existed in the “abstract plane” - outside of the real, physical universe - where the rules constantly changed depending on the context. Example: means the derivative of a function . But then, at some point in school, people started multiplying both sides of an equation by . What the hell? What does that mean? Why was that allowed?

A valid proof if I've ever seen one.

This post is about the most helpful thing I’ve done in my studies: throw out the “abstract plane.” I don’t think it exists. In fact, I think it’s a lazy, non-scientific way of thinking, and getting rid of it has been the only way I’ve been able to distinguish sense from nonsense when talking about math and language.

I’m about to get pretty pedantic about what I mean by “abstract” here, so a little thought experiment might make things clearer.

The Copy-Universe Thought Experiment

Imagine we want to make a simulated copy of our universe. We want our copy-universe to behave exactly like the original, so that in 10 seconds, or 10 days, both the universes are still indistinguishable. What do we need to copy?

Source - The Onion

Well, from high-school chemistry we know that everything in the universe is made out of matter, so we’ll probably need that. So we can copy all the atoms and molecules exactly. You know, the wood molecules stuck together to form your kitchen table, the oxygen atoms bouncing off your skin, etc.

Is that all we need? Well, I suppose we’ll also need to keep track of energy levels, right? How fast are the particles are moving and what direction are they moving in? We’d also probably want to copy light, too. (Or is light matter? Photons are like waves and particles, right? Whatever. )

Source - Stanford

Anyway, here’s the million dollar question:

Is there some “abstract plane” that I also need to copy to my copy-universe?

How would we even do that? We already copied everything we could think of. What’s left? If the abstract plane really never interacts with the physical universe, then it should equally never interact with our simulated copy-universe. So why do we need it?

Definitely knows about triangles. By Justus Sustermans - Link, Public Domain, Link

Consider: if I make a perfect physical copy of Galileo, but I “forget” the abstract plane, it’s not like Galileo won’t know what a triangle is anymore. His brain, which is made out of atoms and neurons, which we copied, will still be exactly the same, and Galileo will do his usual Galileo things. He’ll still be able to walk around, breathe, and talk about triangles. There’s no reason to believe there’s some special property of his brain that would behave differently in the presence of an abstract plane. His brain is just a machine, with wet cogs. It’s just atoms, moving in accordance with the usual laws of physics.1

So the abstract idea of “triangles” doesn’t really live in the abstract plane; it lives in the physical brains / neural circuits of everyone who knows what a triangle is and can name them.

You can tell I’m kind of waving my hands with the physics here. There’s probably all sorts of things I’ve missed (dark matter, quantum effects, etc.). But I’ll leave the quantum theory to the physicists. For my intents and purposes, thinking about brains and stuff, I’ll answer with a resounding no, we don’t have to worry about the abstract plane.

Practical Physicalism

So what’s the alternative? How should we think about abstract things? Ideally, any new idea we encounter should be broken down in terms of our existing model of how the world works. This is first principles thinking, where you maintain a core set of fundamental knowledge and build up all other concepts on top of those. Abstract ideas shouldn’t be an exception, they should be included in our single, unified model of How Things Work. If we let them live in the “abstract plane” where anything goes, we stop being able to apply reason and common-sense to them.

Richard Feynman and Elon Musk love this type of thinking.

But what should that core set be? For me, talking about brains and happiness and stuff, classical physics is good enough. At the end of the day, everything that matters to me can be conceptualized as groups of atoms or the movement of atoms over time. This is very close to Stephen Wolfram’s new kind of science, where everything in the natural world can be viewed as a mechanistic computation. The universe is a machine, whose parts follow some basic rules through time. It’s also very close to the philosophy of physicalism, which asserts that everything is physically manifested.

Physicalism makes some bold claims about everything, which I can’t really support. Instead, I use what I’ve been calling Practical Physicalism: everything that matters to me must be physically manifested. Here’s a simple litmus test for whether something is physically manifested:

Can you touch it, see it, or otherwise sense it? If not, could you theoretically build a device that would let you measure / observe it indirectly? If so, that thing is physically manifested and worth discussing.

Source - Pinterest

It’s a very practical definition, hence the name. And to me, it makes a lot of sense. If something is truly “abstract,” and you can never see it, touch it, or observe it, even indirectly or theoretically… if it never impacts your world, even minutely, then why should you care about it? If something is truly abstract, it’s not worth discussing, because it will never impact any event in your life.

This forces us to define abstract things concretely, in terms of physics, rather than letting them be some ethereal non-existence. Even if the only physical manifestation is some esoteric neural circuit in your brain, that can’t be observed any other way than cracking open your skull and measuring the electrical potentials.

Source - Quantifying the visual concreteness of words and topics in multimodal datasets

It also allows us to start thinking about how concrete or abstract things are. For example, check out this paper on operationalizing and quantifying “abstract-ness” and “concreteness” for specific machine learning datasets that are grounded in pictures. That might help us solve our “happiness” problem.

So I’ve said that this way of thinking has really helped me come to terms with math and language, but I haven’t said how. That’s a longer discussion, so I think I’ll save it for another post.

All models are wrong, but some are useful. - George Box

And yes, all models are wrong. But I think this model of thinking is very useful, especially in a field like machine learning and natural language processing where all we do is think about thinking. Dispelling the “abstract plane” idea early on will benefit everyone, I think.

- Mitchell


  1. What if brains are special? That is, they don’t obey the normal laws of physics, like gravity? Well, then maybe we could call that effect the “abstract plane,” and say that its observable, physical manifestation is to subvert gravity (or electromechanics etc.) to make our brains behave differently than normal physical systems. This would be fine. We can update the physics textbooks. My problem isn’t the existence or non-existence of the abstract plane, but its usage as an excuse to do lazy science / engineering.