[P&N] Chapter 9: Symbiosis

Citation: Edward A. Lee, 2017: Plato and the Nerd - the creative partnership of humans and technology. MIT Press, Cambridge, MA

The Notion of a Continuum

Just as the title of the book suggests, I found that philosophy constitutes a lot of contents of this book. The question that is this world continuous or discrete, for example. I used to consider these questions with no answers not worth our time to even think about, and I realized that it is a big mistake now. In fact, it worth a lot.

Because computers operate entirely within a realm of countable sets, continuums are out of reach for computers.

But do continuums exist in the physical world? No one can actually answer this question yet. We have witnessed the quick development of quantum mechanics these years. We have Planck length, denoted \( \mathcal{l} _ {P} \) and Planck time, denoted \( t_P \). So how do measure the physical world which is smaller than the Planck unit? We can’t, because we need more energy to take that kind of measurement. And if we gather so much energy in such tiny space, we might generate a black hole. The fact is that you want to measure something but end up generating a black hole.

After reading part of this book, I told myself - “If you find something so weird and you can’t get out of it, think about the map and territory.” So what if the question itself is just a “platonic question”? What if it’s asking whether a gray thing is black or a white? What if the world can be both continuous and discrete like wave-particle duality? What if the words like “continuous” and “discrete” are just models? Because all models are wrong, but some of them are useful. Then I guess we can leave the question for a while if we think in this way.

However, the reason that they are useful as models is that these models provide simpler explanations of the physical world than models that reject continuums. Applying the principle of Occam’s razor…

According to the last chapter, why would nature limit itself to the smallest of infinite sets?

The Impossible Becomes Possible

First, let’s examine the concepts of “computable” and “describable”, between which has a really interesting distinction. For example, no computer can give us all the digits of \( \pi \) in finite time, but a computer can give us any arbitrary digit of a decimal or binary representation of \( \pi \). That is to say, we can have a finite program “describes” the infinite number \( \pi \), in which case this infinite sequence is “describable”.

But there are many more real numbers where there is no computer program that can give us any arbitrary digit of the number. … \( \Omega \) is such a number, whose binary representation can be used to solve Turing’s halting problem for a particular binary encoding of Turing machines.

Because we the halting problem is known to be not computable, the number \( \Omega \) is not describable, either.

But what if we have a simple platonic balloon outputs the circumference of a circle given its diameter. Now we can have a perfect representation of \( \pi \) all in once. The problem is that we don’t know the function it realizes, we can’t have the numbers.

In fact, any measurement of the circumference of the balloon is simply putting the output information into another form. What’s wrong with the original form given to me by the balloon?

Lee’s answer is there is nothing wrong with that. In fact, a function does not necessarily need to be describable to be useful. He used the examples of “cars” and “inductors” to illustrate this point. And I think there is more to be done in his illustration to make this point hold.

But let’s skip this point and come to another main point of Lee, which is really remarkable to me:

What makes computers so effective, so useful to humans, is the many possible interpretations we can assign to bit sequences. The partnership of computers with humans is the real source of their power.

Digital Psyche? (Strong AI)

Is it possible for computers to realize humans’ cognitive functions in the future? This question is rather controversy. Lee believes the likelihood is rather remote for the following reason:

1. “Consciousness” is the property of the brain itself which cannot be measured from the outside world.
2. This requires that computers be universal information-processing machines or that the brain be limited to the same class of functions that computers can realize.
3. Passing the Turing test can tell us nothing about whether a computer(or even a human) has consciousness - consider the Chinese room argument.

But if you assert that emulating something in “arbitrarily fine detail” is equivalent to actually achieving that something, then you would have to also assert that no meaningful difference exists between rational numbers and a continuum.

Symbiotic Partnership

There is no question in my mind that humans are coevolving with computers. If computer and software form organisms, then they depend on us for their procreation. We provide the husbandry and serve as midwives. In exchange, we depend on them to manage our systems of finance, commerce, and transportation.

This metaphor seems really interesting to me, so I mark it here for future review and understanding.

Many doomsayers worry about machines enslaving humans someday. But Lee think stronger connection and interdependencies between man and machine could create a more robust ecosystem.

Evolution is a natural process. It is pointless to simply fear it, and if we understand what is happening, we can help guide it in desirable directions. Creating human-like digital psyches is, in my view, not a desirable direction. Fortunately, it is probably not even achievable, at least with today’s computer design. Instead, the real power in the partnership between man and machine comes from their complementarity.

Incompleteness

In order to understand the complementarity, we have to understand the fundamental strengths and limitations of both partners.

Several definitions:

1. Complete: every sentence in the language can be proved true or false.
2. Consistent: no sentence can be proved both true and false.

Consider “Godel’s sentence” next:

“There is no proof for this sentence.”

If we reason through this sentence, we will conclude that no language that can express Godel’s sentence can be both complete and consistent.

In fact, any formal language that is rich enough to describe addition and multiplication of natural numbers can make Godel’s sentence or a sentence logically equivalent to it.

The essence of Godel’s sentence is its self-reference. The sentence talks about itself, reminiscent of the self-scaffolding of software, human consciousness and self-awareness, and Searle’s “the concept that names the phenomenon itself a constituent of the phenomenon.” It suggests that formalisms capable of self-reference are all problematic.

And Hawking points out that such self-reference is also intrinsic in science because the humans who are building models of the physical world are part of that same physical world:

We can our models are both part of the universe we are describing. Thus a physical theory is self-referencing, like in Godel’s theorem. One might therefore expect it to be either inconsistent and incomplete.(Hawking, 2002)

But it’s extremely important to notice that the formal languages considered by Godel only permit us to make a countable number of mathematical sentences. Moreover, his incompleteness result only applies to formal languages that are rich enough to describe arithmetic on natural numbers - another countable set.

If instead we look at formal languages that describe arithmetic on real numbers, then the theorems do not apply. Tarski showed in 1948 that a natural theory of real numbers that expresses addition and multiplication, the so-called theory of real closed fields(RCFs), is both complete and consistent(and also decidable, an even stronger property which asserts that the truth or falsehood of any statement can be determined by an effectively computable function.)

It may confuse you a lot, isn’t \( \mathbb{N} \in \mathbb{R} \)? Why the hell the above statement hold? Here is the tricky point: you know that \( \mathbb{N} \) is a subset of \( \mathbb{R} \) from primary school mathematics but RCF isn’t what you learned before. Within RCF it is nonsensical to talk about the natural numbers because there is no way to even say what they are.(Credit goes to Meefims) So the conclusion is that we can not define natural numbers using RSFs, which is to say, there is no formula \( F(x) \) of our language such that \( F(a) \) is true in the reals if and only if \( a \) is a natural number.