This post is part 2 of my attempt to understand the discussion of Godel’s incompleteness theorem presented in The Fabric of Reality by David Deutsch (DD). See Part 1 here.
Below is a summary of what I thought the relevant parts of Chapter 10 were. I included a big chunk of the beginning of the chapter cuz it seemed like relevant context to understand the problem situation Godel was facing.
David looks back on his discussion of Cantgotu environments and asks:
I have said that there exist infinitely many [Cantgotu environments] for every environment that can be rendered. But what does it mean to say that such environments ‘exist’? If they do not exist in reality, or even in virtual reality, where do they exist?
DD asks whether abstract, non-physical entities like numbers & laws of physics exist. He wants to distinguish things that have an independent existence vs. things that are features of our culture, arbitrary, etc.
DD says if we want to know whether a given abstraction exists, we should ask whether it “kicks back” in a complex, autonomous way. He talks about natural numbers as an example. We came up with them as an abstract way of expressing “successive amounts of a discrete quantity.” But after we made them, it turns out they have all these properties we have to figure out (like characteristics of the distribution of prime numbers.)
Since we cannot understand them either as being part of ourselves or as being part of something else that we already understand, but we can understand them as independent entities, we must conclude that they are real, independent entities.
DD says abstract entities are intangible and don’t literally kick back. He says proofs play the role in maths experiment and observation plays in science, and says that mathematicians are big on thinking proofs are absolutely certain.
DD talks some about Pythagoras and Plato. Pythagoras thought regularities in nature were expressions of mathematical relationships in natural numbers. Plato thought the physical world wasn’t real and thought things we experience in our world were reflections from the world of Forms (the Forms including numbers like 1,2,3, mathematical operations etc.) DD also mentions Plato’s theory of in-born knowledge.
DD says that mathematicians agree “mathematical intuition” is a source of absolute certainty, but then disagree about what mathematical intuition tells them, heh. He gives the example of imaginary numbers. There were proofs about real numbers that involved imaginary numbers, and some mathematicians objected because they said imaginary numbers weren’t real. Good example.
DD talks about a crisis in mathematics. Aristotle had figured out the laws of logic and syllogisms, and said that all valid proofs could be expressed as syllogisms. He hadn’t proved this though. And DD says the crisis was modern mathematical proofs weren’t expressed syllogistically and involved tools outside the classical logic rules. So this proved that Aristotle’s rules were inadequate. And people were worried that maybe the new stuff wasn’t absolutely certain.
DD talks about responses to this crisis. One was intuitionism, which tries to construct intuition in a super narrow way only based on supposedly unchallengeable, self-evident aspects. This leads to them denying infinite sets.
DD says intuitionism had some value (like inductivism) in that it dared to question some received certainties. But it ultimately involved retreating into an inner world/domain that actually makes explanation of even that inner domain harder. E.g. intuitionists deny infinities. So there must be finite natural numbers. How many? And then why can’t you form an intuition about the next natural number above that one? Intuitionists reply to this by denying logic, specifically the law of the excluded middle, which says there’s no third possibility between a given proposition and its negation.
…by severing the link between their version of the abstract ‘natural numbers’ and the intuitions that those numbers were originally intended to formalize, intuitionists have also denied themselves the usual explanatory structure through which natural numbers are understood. This raises a problem for anyone who prefers explanations to unexplained complications. Instead of solving that problem by providing an alternative or deeper explanatory structure for the natural numbers, intuitionism does exactly what the Inquisition did, and what solipsists do: it retreats still further from explanation. It introduces further unexplained complications (in this case the denial of the law of the excluded middle) whose only purpose is to allow intuitionists to behave as if their opponents’ explanation were true, while drawing no conclusions about reality from this.
DD then turns to Godel:
Thirty-one years later, Kurt Godel revolutionized proof theory with a root-and- branch refutation from which the mathematical and philosophical worlds are still reeling: […] Godel proved first that any set of rules of inference that is capable of correctly validating even the proofs of ordinary arithmetic could never validate a proof of its own consistency. […] Second, Godel proved that if a set of rules of inference in some (sufficiently rich) branch of mathematics is consistent (whether provably so or not), then within that branch of mathematics there must exist valid methods of proof that those rules fail to designate as valid. This is called Godel’s incompleteness theorem. To prove his theorems, Godel used a remarkable extension of the Cantor ‘diagonal argument’ that I mentioned in Chapter 6. He began by considering any consistent set of rules of inference. Then he showed how to construct a proposition which could neither be proved nor disproved under those rules. Then he proved that that proposition would be true.
So basically Godel showed that we can have no fixed way of knowing if a mathematical proposition is true.
Skipping ahead some in the chapter…
DD says one of Godel’s assumptions was that a proof can have only a finite number of steps. DD says this is correct as far as we know according to quantum theory.
DD says Godel inherited Greek conception that a proof was a particular type of object (a sequence of statements that obey rules of inference.) But really it’s better thought of as a process. He says that with the “classical theory of proof or computation” this distinction isn’t hugely significant because you can just make a record of the process and therefore have a proof “object”. But with quantum computer calculations lots of stuff is happening in other universes so you can’t record all that stuff. And this shows that the old traditional mathematical method of trying to have totally certain stuff by stripping away every source of ambiguity and error isn’t viable.
In the next part I’ll try and sum up.