What is Symmetry

Here is a quick, painless introduction to the kind of symmetry that you probably learned in school. In that lesson, we learn that there are different kinds of symmetry:

• Rotational
• Translational
• Mirror

and so on. What this video (and your teacher) did not explain was what symmetry is in general. If you learned about symmetry in University Math, your idea was expanded a bit to come closer to what symmetry is in general. The first step is to think of symmetry as a set of transformations (rotate, move, flip) rather than a property of whatever it is that is being transformed.

And that's just the start. For example, in physics, you learned the Lorentz Transformation is a symmetry group in the "real world" that leaves the laws of physics unchanged. Newton's laws are not symmetric in this sense but with a bit of tweaking (Special Relativity) they can be fixed.

In physics, the Lorentz transformation (or transformations) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other.

Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating in curved paths, rotational motion with constant angular velocity, etc.). The term "Lorentz transformations" only refers to transformations between inertial frames, usually in the context of special relativity.

Special relativity boils down to "fixing" the laws of physics so that they apply "locally" in non-inertial frames of reference.  I promise: no more about General Relativity.

Lorentz transformations are a special case of Gauge Symmetry, which is a concept used in the "Core Theory" of Quantum Mechanics and gets us finally to the point where we are talking about "real" symmetry in the "real" world. You need it to understand quarks for example.

This can all get pretty abstract and make your head hurt if you follow it all the way to Quantum Mechanics. It's best to stand back and ask what symmetry is. Wilczek describes it briefly as "change without change".  Or perhaps we could say you can change one thing that doesn't matter and everything that does matter is still the same.

In this blog we will be interested in a few kids of symmetry that aren't covered in school and are perhaps just a bit more general than even Gauge Symmetry. These involve transformations between domains I, M and R and within those domains or within sub-domains. For example the relationship between two ideas that are "like" each other (within the M-domain) is seen as a symmetry transformation. Some transformations are "better" than others and the criterion is symmetry. Does the transformation leave the idea the "same" in important aspects. This is similar to Hofstadter's idea of the "Essence" that is preserved in a "good" analogy.

Human beings seem to love all kinds of symmetry for good reason - they help economize on "representation" of "things" in the brain. Wylczek draws an interesting parallel with data compression. The Core Theory of Quantum Mechanics, which more or less represents everything we know about the physical world, can be written down in a few dozen strange looking characters, yet everything in human experience is a special case of it. Wylcze's treatment of this issue inspires this whole blog since he wrote a whole book about it: "A Beautiful Question" which basically points out the astonishing symmetry between our ideas (especially the Core Theory) and the real world. The equations of the Core Theory are incredibly symmetric and the real world seems to "like" symmetry a lot too. The "Beautiful Question" is a response to this situation. You are left with astonishment and wonder: a question rather than a "theory".

Letter to John Heerema, Sept 24, 2016

I am particularly attracted to the idea of the "fractal", which is a scale-independent repetition of a theme. Nature seems to "discover" a few patterns then use them in endless patterns and variations. This is one of the building blocks of what is taking shape as a "theory of mind". To build a "super mind", it would make sense to design the largest scale of this mind in the cloud to tap the computational power of millions of volunteer machines (as in the cloud project to classify galaxies).

However, perhaps this as already been done. At least to serve in my "thought experiment", Google may be enough. The "theme" that has me fascinated is the mapping of one brain state into another, using something similar to the Shrodinger Wave Equation. That equation describes the state of the universe in the "next instant" as a function of the state of the universe in the "previous instant". The mental equivalent would be the set of memes arising from the set of memes in the previous instant. At the level of the stream of consciousness of a single person, this might be a good way describing "thinking". There are good reasons to suspect that a single human brain can sustain several of these steams at once, perhaps millions. Daniel Dennett likes to use the metaphor of thousands of mini-minds "voting" on what should be brought to attention at any one moment. I think he's perhaps more right than he thinks. It's not just a metaphor. of On a smaller scale, this is a model of perception (what things "look like" or "sound like") and also a model of overt action ("why" we do some specific thing). 

Beyond the level of the individual, memes continue to map. In physics, we speak of a "core" theory, which is the Shrodinger Equation restricted to a specific domain, within which its results are "for all intents and purposes" accurate (for example, in some situations, you can ignore gravity). It's not hard to imagine the meme mapping that's going on in one person's stream of consciousness, but there seems to be no reason to restrict the idea to just one person or the lifetime of one brain. Entire societies "think" in this way, producing a new set of memes from the preceding ones every second. 

I'm fascinated by history. When I learn about World War I or the career of Napoleon, it seems to me that the entire world was "thinking" - following one set of memes to another in a crazy way that makes sense on some level. You can' speak intelligibly about these things using physics. They are more like thought processes and that's the way they are usually described.

What does Google do? You input a meme, in the form of a word, a question or a collection of words that, according to your best efforts, describe an idea. Google pops up with a few million "related" ideas ranked in order of "best guess". How is that different from thinking?

All this can seem interesting but vague. Dawkin's original idea of the meme as the mental equivalent of a gene didn't catch on because there was no way of defining what "meme" means. What is an idea? What is a perception? Like fundamentally new ideas, the word "meme" needed to be defined by using it. Hofstadter is the master when it comes to showing us how to use it.

Hofstadter is quite familiar with the word "meme" but talks about it using the more familiar term "category". His work can be seen as an effort to describe what we mean by a "category", and it turns out to have all the recursive, fractal, self-referencing properties of thought itself. In fact, Hofstadter seems to claim that thought is "nothing more" than forming categories.

Hofstader's work lends the requisite elegance to "meme" theory. He thinks (and he may be right) that meme mapping completely accounts for consciousness. Once you "grok" Hofstader's way of thinking about thinking, lots of the literature on the mind seems to suddenly make sense. I'm currently reading Gilbert Ryle's classic "The Concept of Mind" (1947) which is one of the best-known assaults on the Cartesian theory of mind. Ryle sits right at the edge of a revolution. He's trying to teach us how to speak sensibly about thought processes as being in the same category as "overt action". Seen through Hofstader lenses, his argument convinces me that actions are memes. AHA! 

To my mind, these insight is at least as productive as Newton's discovery of calculus. Applying meme theory to its own emergence, we see that it's not the product of a single mind. It's popping into existence all over the place. Like calculus and relativity, it was there all along, waiting to be discovered. I'm just running with it to see where it leads

P.S.

After sending this note off to John, I let my dogs in from outside. I watched the two of them "calculate" how to get in the door while dealing with the distraction posed by he competition between them. It struck me that dogs (and animals in general, even fleas) map perception into action more or less directly ("instinctively"). That is, meme theory applies to everything with a brain, no matter how simple that brain may be. This is thoroughly consistent with the fractal nature of mind.

Idea #2 - how does Dawkin's original idea - survival of memes - work in to all of this? Can we make sense of the "survival" idea with respect to memes?