I came across this book after a short exchange of emails about books concerning mathematical philosophy. Douglas Hofstadter is known for being the writer of Gödel, Escher, Bach: an Eternal Golden Braid, an excellent book from 1979.
The author defines this book as his best attempt at describing what “the human condition” is. The prologue is a dialogue between Plato and Socrates about consciousness. The former suspects that consciousness is an illusion, while the latter believes in consciousness’s reality. The two interrogate themselves on the nature of thinking or the meanings of saying “I know” and “I am alive”. They conclude that these sentences are indeed not so obvious to understand and that we – humans – think they arise through conscious thought, yet, that thought is seen to be an automatic reflex.
To introduce the topic of consciousness, Hofstadter draws a “consciousness cone” in which he places the small entities – between atoms and mosquitoes – at the bottom and humans at the top. The entities at the bottom have little or no consciousness, living creatures listed in the middle, such as chickens or goldfishes, have less (but some) consciousness, brain-damaged and senile humans are one step below the “normal adult humans” with “lots of consciousness”. The concept of “soul” or “I” is described as a blurry numerical variable that ranges continuously across species. This view overcomes the binary distinctions supported by philosophers of mind between “possessing intentionality” – i.e., having beliefs and desires, and the “mere” ability to juggle meaningless tokens in complicated patterns.
The author refers to mathematics as something primarily related to reasoning. Gödel’s Proof and paradoxical sentences like “I am lying” are examples of the circularity of self-referential phenomena. “Video voyages”, originated by pointing a video camera towards its screen, become ways to grasp the mysterious matter of endless corridors and black holes. Despite the richness of the visual feedback, the video system lacks a repertoire of symbols that can be selectively triggered, thus cannot be a valid representation of perception. Perception implies that some kind of input of a vast number of tiny signals eventually ends up being processed by a small subset of a large repertoire of latent discrete structures with representational quality – i.e., symbols. On this subject, perception in mammals and robots differs because robots, like mosquitos, suffer from an abstract symbols deficit.
However, humans’ abstraction abilities can encounter concrete walls, using the words of the author:
“We build up an intricate, interlocked set of beliefs as to what exists “out there” – and then, once again, that set of beliefs folds back, inevitably and seamlessly, to apply to our selves.”
Humans’ sense of absolute certitude about abstract things first depends on the reliability of internal symbols to directly mirror the concrete environment, and second concerns the reliability of thinking mechanisms to tell about more abstract entities that cannot be directly perceived.
“All of this more abstract stuff is rooted in the constant reinforcement, moment by moment, of the symbols that are haphazardly triggered out of dormancy by events in the world that we perceive first-hand.”
The book thesis about “I-ness” as a strange loop arises from this idea that we are all egocentric, and what is realest to each of us, in the end, is ourself.
The author defines strange loops as paradoxical level-crossing feedback loops. Principia Mathematica serves as an introduction for pattern-hunting and proofs-searching at the basis of mathematicians’ constitutional curiosity towards concepts that do not, a priori, seem related at all. The direct link with reasoning comes from the Mathematician’s Credo that where there’s a pattern, there’s a reason. This mindset bring Gödel to the idea that all sorts of infinite classes of numbers can be defined through various kinds of recursive rules.
Hofstadter connects Gödel’s austere mathematical strange loop to the very human notion of a conscious self.
“You make decisions, take actions, affect the world, receive feedback, incorporate it into yourself, then the updated “you” make more decisions, and so forth, round and round. It’s a loop, no doubt – but where’s the paradoxical quality that I’ve been saying is a sine qua non for strange loopiness?”
The conclusion of the book is again a dialogue. A believer and a doubter discuss the ideas of I Am a Strange Loop. One of the underlying meanings is that much of our life is incredibly random, and we have no control over it. We can will away all we want, but much of the time our will is frustrated.
Below some extracts from the book, and a bunch of other suggestions.
“But our glory as human beings is that, thanks to being beings with brains complicated enough to allow us to have friends and to feel love, we get the bonus of experiencing the vast world around us, which is to say, we get consciousness.”
“You and I are mirages who perceive themselves, and the sole magical machinery behind the scenes is perception – the triggering, by huge flows of raw data, of a tiny set of symbols that stand for abstract regularities in the world. When perception at arbitrarily high levels of abstraction enters the world of physics and when feedback loops galore come into play, then “which” eventually turns into “who”. What would once have been brusquely labeled “mechanical” and reflexively discarded as a candidate for consciousness has to be reconsidered. We human beings are macroscopic structures in the universe whose laws reside at a microscopic level; As survival-seeking beings, we are driven to seek efficient explanations that make reference only to entities at our own level.”
“Our very nature is such as to prevent us from fully understanding its very nature. […] We humans beings […] are unpredictable self-writing poems.”
Other books on mathematical philosophy and scientific method:
- Bertrand Russell – Introduction to Mathematical Philosophy
- Karl Popper – Logic of Scientific Discovery, The World of Parmenides: Essays on the Presocratic Enlightenment, The Two Fundamental Problems of the Theory of Knowledge
- David Deutsch – The Beginning of Infinity: Explanations That Transform the World
- Thomas S. Kuhn – The Structure of Scientific Revolutions
The books’ prompter asked me to which of the two following group I belong:
- the one who believes that all ideas exist and we rediscover them, thus mathematics exists without us;
- the one who assumes we really create and invent, thus mathematics is our invention.
In this regard, I found a divulgative TED talk from Jeff Dekofsky. What’s your opinion on that?