It’s been a great week on the blog with thoughtful and thought-provoking comments cropping up everywhere. Several threads have touched on the question phrased most succinctly by Al. V. on the Unreasonable Effectiveness of Physics thread:
Are the laws of mathematics inherent in our universe, and therefore really a product of physics (and not the other way around), or are they supra-universal?
This question, of course, plays a starring role in The Big Questions , where I’ve explained why I believe that the supra-universality of mathematics (thanks for that word, Al!) gives the most coherent explanation of why anything exists at all.
The same issue arose—with many comments well worth rereading—on the What Are You Surest Of? thread. There, Bill T. raises a provocative question: If (as I’ve suggested in The Big Questions), the physical world is in some sense “made of mathematics”, why can’t we take this one step further and speculate that mathematics is “made of logic”? The answer is that Godel’s theorems make that a very difficult step to take. Starting with the standard axioms of arithmetic, and armed with the full power of logic, there remain true statements about arithmetic that cannot be proven. This means that arithmetic must be more than just logic.
(Logicians in the audience will want to quibble about what constitutes the “full power of logic”; I will duck that question to avoid a long technical digression right now.)
I offered the first in a planned series of posts on attempting to see Darwin through 19th century eyes; Snorri Godhi checked my math carefully and made a valuable correction. And we got into a debate about whether cellphones really would have resolved the plot tension in a substantial fraction of 20th century movies.
Thanks for the fun and the enlightenment. I’ll be back on Monday with more.
Years ago, I managed to persuade myself that Godel’s theorems need not shake me out of my nominalism. Let’s see if I can reconstruct my reasoning. A good place to start:
Starting with the standard axioms of arithmetic, and armed with the full power of logic, there remain true statements about arithmetic that cannot be proven.
[a] Those true statements cannot be proven by logic plus the standard axioms of arithmetic, but they _can_ be proven, otherwise we would not know that they are true.
[b] One such “true” statement is the statement “this statement cannot be proven by standard arithmetic”. But this is true if, and only if, arithmetic is self-consistent; so we move on to
[c] another such “true” statement: “arithmetic is self-consistent”. To prove this, we need another, more powerful axiomatic system X (and logic). But we cannot prove that X is self-consistent, and therefore the proof is not conclusive.
From [a], [b], and [c] I conclude that truth is, for most practical purposes, the same as provability (although truth within an axiomatic system is not always the same as provability within said system). But only for practical purposes: I believe that the self-consistency of an axiomatic system is an objective fact, but a fact that we cannot know.
Thank you, Steven, for responding to my question and referring me to Godel’s theorem. I think I’ve gotten closer, but not fully there.
Would you say that the “incomplete” set of mathematical axioms (from Godel) is of the first degree of infinity? It would seem so to me.
If that first degree infinite set of axioms represents all of mathematics, then the set of truths that “couldn’t have been different” (and thus couldn’t have been designed) is of the first degree infinity.
In the physical world, there are sets of truths of higher degrees of infinity. Therefore the physical world is more complex than mathematics could ever be.
I guess for me to change my mind I’d have to understand that either the set of mathematical axioms is “larger” than first degree infinite set… or that there doesn’t really exist anything in reality more complex (“larger”) than a first degree infinite set.
WRT my comment above: I wrote it because I noticed that Steven and some other people reading this blog know more about Godel’s theorem than I do (which is not difficult), so I thought this is a good chance to try out my understanding of it. Sorry for not waiting to read the book.
And thank you Steven for paying attention to our comments. I have not yet re-checked your derivations from the other day, but I will.
Bill T.: If I understand your question correctly, you’ll find it addressed in the footnote on page 27 of The Big Questions. The issue is not about the sizes of the set of truth and the set of axioms. Instead it’s this:
There are many true statements about arithmetic. You are allowed to pick as many of those as you want (even infinitely many) and call them axioms. There’s just one rule, though: You have to give an explicit procedure that will let me distinguish an axiom from a non-axiom in a finite amount of time. One way to do that is to restrict yourself to finitely many axioms. That is not what we usually do, though. Instead, we have infinitely many axioms, but they all fit a certain pattern.
Here’s an axiom: If there are any numbers equal to their own cubes, then there is a smallest one. Here’s another: If there are any even prime numbers, then there’s a smallest one. Here’s another: If there are any numbers not equal to themselves, then there’s a smallest one. Et cetera. Any statement fitting that pattern is an axiom, and this set of axioms obeys the rules because it’s easy to check whether a given statement fits the pattern or not.
Now if you don’t like that set of axioms, you’re welcome to pick another. The only rules are: Your axioms must all be true, and they must be recognizable. That means you’re not allowed to say “I take all true statements as my axioms”. That’ s not a legitimate set of axioms, because I have no finite procedure for recognizing whether a given statement is true or not.
Now Godel’s theorem (in part) says this: You pick your favorite set of axioms. I’ll always be able to find a true statement that does not follow from them.
Thank you, but my book (free press hard cover) doesn’t have a footnote on page 27 so I’ll try rephrase…
I may have been using my terms improperly.
Specifically, would you agree or disagree with each of these statements:
1. The complexity of a system can be measured by the size of the “data” required to fully describe it.
2. The entirety of what we call mathematics is an infinite enumerable set of truths (or axioms). So the size of the data (complexity) required to describe mathematics is equivalent to the size of data in a first degree infinite set.
3. The size of the data (complexity) of reality is larger than the data in a first degree infinite set. (Time adds a continuum of truths that are innumerable).
4. Therefore reality is more complex than mathematics.
Sorry; I think I meant page 97, though I don’t have the book in front of me at the moment to check.
In your point 2), you are right that the set of true statements about arithmetic is enumerable. But it is not *recursively* enumerable. In other words, there exists an enumeration of those true statements, but there is no procedure to produce that enumeration. Your formulation fails to notice this distinction.
In your point 4), you are assuming that because “reality” is uncountable it can’t be described by countably much data. But that is false. The real numbers are uncountable, but all true first-order statements about the real numbers can be derived from a countable set of axioms. By contrast, the natural numbers are countable, but no countable set of axioms suffices to imply all true statements about them.
Thank you Steve,
I jumped ahead in the book and read quite a bit more.
I think I’ve pinpointed specifically where we differ. My belief that reality is not able to be described by countably much data stems from my not believing that Determinism is true. Given Determinism is true, I take your point.
Also, I guess since I believe in free will (as you do), but reject determinism (quantum physics and Schrödinger’s cat make me doubt it) I’d be considered a Philosophical Libertarian (nicely rounding out my Political Libertarian side :) )
I’ll have to read up on the Compatibilist philosophers of the day to get greater insight.
On the determinism note, I recommend you checking out “Quantum Enigma” by Bruce Rosenblum and Fred Kuttner http://quantumenigma.com/. They provide some intriguing insights to how physics meets consciousness.
Thanks again for responding to your readers.
Bill: Our difference is not over whether reality can be described by countably much data. It is over whether the natural numbers can be described by countably many data. It is TRUE that that the set of true statements about the natural numbers is countable. It is FALSE that this set of true statements can be described in any form that a human being (or a computing device) would be able to recognize.
Here’s an axiom: If there are any numbers equal to their own cubes, then there is a smallest one. Here’s another: If there are any even prime numbers, then there’s a smallest one. Here’s another: If there are any numbers not equal to themselves, then there’s a smallest one. Et cetera.
Sorry, but I am a bit confused: how is this different from having just one axiom that says: “for any set of numbers which fits pattern X, there is a smallest number”? (I am not sure what the pattern is: there are too few examples for a guess.)
More generally, if there is an algorithm that can be used to validate an axiom, then is this algorithm not the “real” axiom, with the axioms being demoted to the level of theorems?
Apologies if this is all in the book: even if I ordered it, it would take weeks to get it. But I’ll take “it’s all in the book” as an answer.
Sorry, but I am a bit confused: how is this different from having just one axiom that says: “for any set of numbers which fits pattern X, there is a smallest number”?
The formal language of first-order logic allows you to say “for any number” but does not allow you to say “for any set”.
In general, your axioms have to be about numbers, not about sets of numbers.