Nevit Dilmen, CC BY-SA 3.0, via Wikimedia Commons
I spent much of today re-reading David Deutsch’s excellent book The Beginning of Infinity and making detailed notes. (Naval wrote a good overview of the book.)
It’s an insanely ambitious book, and I ended up spending several hours reading, taking notes, and thinking intensely, as I worked out the implications of the ideas contained within, and tried to figure out the connections to various other theories or fields of interest.
Deutsch’s core thesis is that the future growth of knowledge, and therefore progress, is potentially infinite, and so we are always effectively at the beginning of infinity.
Any finite number compared to infinity is effectively zero; even if humanity or our descendants — humans, posthumans, AIs, or hybrids of some kind — survive for a trillion years, they will still be at the beginning of infinity (says Deutsch), and they will have an unlimited number of ideas to discover, problems to solve, and improvements to make.
I won’t post all of my notes here, but I’ll likely continue reading and thinking about that book and others in a similar vein over the next few weeks, and writing out my own conclusions as they solidify.
Deutsch writes at great length about knowledge discovery, and the different means by which knowledge is instantiated in physical reality.
One of his core theses is that there’s a fundamental similarity between knowledge discovered by human beings, and the ‘knowledge’ discovered by biological evolution — for example, he claims that the design of complex biological organs, such as eyes, represents ‘knowledge’ encoded in DNA. I don’t fully agree — I’d posit that these are two very different kinds of things, and that the latter is only similar to the former by analogy.
Nevertheless, it’s a fruitful premise to consider, and his focus on explaining the ‘implicit knowledge’ represented by biological ‘design’ leads to a worthwhile discussion.
I thought his discussion on the historical development of theories of biological evolution was very well-structured, and identifying the conceptual flaws in the various historical explanations for biological complexity — creationism, spontaneous generation, and Lamarckism — was an extremely valuable exercise.
He doesn’t just show how these theories were refuted by the evidence, he explains at length how they’re simply bad explanations — they all fail to actually explain the phenomena in question.
Briefly, all three flawed theories attempt to explain the apparent appearance of design in biological organisms via some other (equally complex) mechanism that is itself left unexplained. (I’m oversimplifying his arguments, of course, but I can’t fully do them justice in a few paragraphs.) The Neo-Darwinian theory, in contrast — natural selection of mutating replicators — does explain the appearance of complexity.
Thinking through Deutsch’s concept of explanation led me to think about how one would explain explanations, which then led me to consider the complex tangle of meta-questions that arises when one considers the mind observing itself. As explanation is closely related to understanding — to properly understand something is to know how to explain it — I will focus on the idea of understanding, and of self-understanding in particular.
It’s possible to get a handle on the idea of self-understanding by first considering simple examples — what does the mind do when it attempts to understand a simple mechanism, like a kitchen tap? — followed by moderately complex examples — what does the mind do when it attempts to understand a complex mechanism, such as a CPU?
Finally, we can ask the really hard question — what does the mind do when it attempts to understand itself? — or even more specifically, what does the mind do when it attempts to understand how it understands? — which has the curious property that the question itself is an instance of the phenomena being explored. This leads to the often-observed “hall of mirrors” effect that arises when questions relating to self-reference, or to consciousness observing itself, are contemplated.
This effect is itself something that the mind can attempt to understand — and this is where the phenomena gets very confusing. It’s easy to dismiss questions involving self-reference as mere mathematical curiosities that ultimately end up in logical paradoxes, and are therefore either contradictory or meaningless. However, the human mind is in fact often capable of identifying logical paradoxes — and this fact then gives rise to even more complex mental phenomena that require explanations.
Below I give an extended discussion on logical paradoxes. Be warned, understanding these ideas will require some extremely heavy conceptual lifting, but there is a point to all of this.
If we consider a very simple logical paradox — such as the famous Liar Paradox: [this statement is false] — we can quickly conclude that the paradox ‘works’ by confusing the means by which we normally assign truth-values to statements.
In the general case we decide whether a statement — such as [this cup is white] — is true or false by reference to physical reality (this is known as the correspondence theory of truth).
The Liar refers only to its own truth-value — or, in other words, it refers only to its own reference — and so it cannot be put into any consistent correspondence with physical reality. (Again, I’m simplifying a complex issue in a few sentences.)
However, we do manage to reason about reality without our minds becoming paralysed by logical paradoxes, which is a clear sign that we have some means to ‘contain’ paradoxes from taking over our thoughts. How do we do this?
To tie this back to the original question — for the mind to truly understand itself, it does indeed need to understand logical paradoxes, how they work, and the mechanisms by which it ‘contains’ them from its regular thinking.
The fact that we can consistently identify logical paradoxes, and that we can often explain how they work, suggests that they represent some kind of consistent phenomena (more specifically, they involve some consistent pattern in the way our minds work). In the same vein, optical illusions — such as the circles in the image below, which appear to be rotating, even though they are in fact static — reveal something about the workings of our perceptual apparatus.
Our minds make sense of reality by means of various mechanisms that translate sensory inputs into a coherent image of the world.
(This does not imply the popular — but wrong — notion that our awareness of the world is an illusion. We can observe that there is a fundamental consistency behind our experiences, and can learn many things about the structure of physical reality by making repeated observations over time. There is a very strong case to be made that “all experience is illusory” is simply a conceptually incoherent notion. For more on this, see the work of James J. Gibson.)
However, the mechanisms by which the mind makes sense of reality can be hacked. The optical illusion above hacks our perceptual apparatus; in a similar manner, logical paradoxes like the Liar hack our conceptual apparatus. Curiously, though, it is possible to understand the mechanisms behind these hacks, and doing so sheds extremely valuable insights into the operations of our sensory apparatus and our minds.
A further thought: if we can explain some logical paradoxes, we can then consider the idea of explainable logical paradoxes; and if we do that we can then construct extremely confusing higher-order paradoxes. One example: [this sentence does not contain a logical paradox that can be explained].
(If you think the bolded statement is true or false, can you prove it?)
Finally: it seems plausible that, no matter how many logical paradoxes you can explain, you can take all of your paradox-explanation methods and use them to construct a new paradox that cannot be explained by those methods. If you figure out how to explain that paradox, then you can construct another new paradox that can’t be explained by your new methods. (If you want to try to connect this idea to Godel’s Incompleteness Theorems, go ahead.) Might there be an infinity of inexplicable logical paradoxes — indeed, an infinite hierarchy of logical paradoxes sorted by classes of inexplicability? Or would that, too, eventually descend into sheer incomprehensibility?
The question I posed above was: what does the mind do when it attempts to understand how it understands?
What was the purpose of this extended discussion on logical paradoxes? Contemplating logical paradoxes — in particular, the issue of explainable logical paradoxes — reveals many insights into the exact borderlines of self-understanding. In particular, pushing one’s understanding of the explicability of logical paradoxes to its limit, is itself a perfect model of pushing one’s self-understanding to its limit.
And finally: your mental activity in reading this article — and mine in writing it — are themselves perfect examples of the phenomena we were seeking to understand.
If you are confused by the issues in this article — which essentially explores the idea of maximally-confusing problems in depth — attempting to resolve the confusions should give you a window into how your mind seeks understanding in general.
Fortunately, it is possible to understand understanding at a reasonable level using more everyday examples, as I will discuss in a future post.