In yesterday’s post, I claimed to have refuted Richard Dawkins’s claim that everything complex must have emerged from something simple by citing the natural numbers, which are provably highly complex (in a very precise sense) yet did not emerge from something simple. Numerous commenters suggested that I’d been unfair to Dawkins, because he’d surely meant his claim to apply only to biological processes.
Here is a quote from Dawkins’s book “The God Delusion”:
Time and again, my theologian friends returned to the point that there had to be a reason why there is something rather than nothing. There must have been a first cause of everything, and we might as well give it the name God. Yes, I siaid, but it must have been simple and therefore, whatever else we call it, God is not an appropriate name….The first cause that we seek must have been the simple basis for a self-bootstrapping crane which eventually raised the world as we know it into its present existence.
I could provide additional quotes, but this one should suffice. Dawkins believes, unless I have misunderstood him completely, that he has a quite general argument, not tied in any way to biology (because the above quote, for example, has nothing to do with biology) to establish that complex structures must have simple causes. That argument, whatever it might be, cannot be correct because the natural numbers stand as a counterexample.
If Dawkins, or any of his defenders, wants to respond that his argument is not intended to apply to the natural numbers, it becomes incumbent on them to point to a hypothesis which is actually used in the argument which would rule out such an application. Absent such a hypothesis, the argument must be erroneous.
I claim to have explained in The Big Questions exactly how the first cause of our Universe could be a mathematical structure that is far more complex than the Universe itself; of course others, like Max Tegmark, have demonstrated this possibility in far more detail than I have. Whether or not Tegmark and I are correct in our beliefs, I claim we’ve at least demonstrated that (as far as we can tell) those beliefs could be true, which, once again, refutes Dawkins’s position.
An argument that leads to flat-out wrong conclusions cannot be a correct argument, even if some of its implications turn out to be true. So I stand by what I said both yesterday and in The Big Questions : Dawkins’s position fails for exactly the same reason that Michael Behe’s does — we have an explicit example that shows that complexity requires neither a simpler antecedent nor a designer.
I suspect that Dawkins (and maybe Behe) would say that the integers are not complex. The complexity is only in the computational complexity of certain formulas in Peano arithmetic. So the integers have simple causes, and the complexity is added later as one makes various tricky mathematical definitions. So where is the contradiction with Dawkins?
#1. I am sure that is an argument. the fact that the natural numbers are complex “in a very precise sense” may mean that the complexity that Dawkins is arguing for is a slightly different sense. I think we have to nail exactly what complexity is first. I am also sure that previous posts have dealt with this issue – any pointers to old posts we can go to? Or a quick summary?
Aren’t the Peano axioms (or any equivalent) a “simple basis for a self-bootstrapping crane which raised the integers as we know them into their present (and eternal?) existence.”?
Well, I have several thoughts. . . .
1.) Existence vs Nonexistence?
This may be a very simple question, but when we say something exists (E.G., the natural numbers) what properties are we attributing to the natural numbers, exactly?
Rephrased: what is the definition of existence? How can you delineate between something that is, and is not? What property indicates “is-ness”?
I think if you can answer the aforementioned question(s), you can then begin to draw a line somewhere in the sand that would demarcate where the hypothesis you ask for begins and ends. But, by that same note, if you can’t point to something that would make the real numbers different from a snail (insofar as existing) we’re in bad shape.
2.) Aren’t Numbers Just Concepts?
I suspect that this is elementary, and this question alone disqualifies me from participating on this blog. But, nevertheless, I have the notion that numbers like 1 are symbols for concepts; you can’t be a 1, but you can count to 1, and you can see that there is 1 of something. And inasmuch as we can conceive the number 1, we can also fathom pretty much anything. And if mere thought is a decided requisite of existence. . . we’re in bad shape.
Moreover, if numbers are concepts of the mind, and perhaps didn’t (don’t) presage the mind, couldn’t it be said that we created them? Something tells me no. But I asked anyways. . . . But, if no, you’d have to agree that we discovered the natural numbers (which is a strange supposition to me).
3.) Complexity?
I heard Professor Landsburg once comment (somewhere) that numbers make-up the universe, and numbers [which explain the universe] are more complex than the universe. I don’t understand how that denotes complexity. Couldn’t that be used as a counterexample to the fact they’re concepts so simple in nature that they can be applied anywhere?
I may be more confused than I know.
“must have been the simple basis for a self-bootstrapping crane ”
Are the integers a self-bootstrapping crane?
Dawkins is clearly referring to structures. Biological structures are the most complex of those but he also seems (I concede) to be including nitrogen atoms, galaxies, eand other structures.
Banging on a bit on the same old drum, from Wiki: “When Dembski’s mathematical claims on specific complexity are interpreted to make them meaningful and conform to minimal standards of mathematical usage, they usually turn out to be false.[citation needed] Dembski often sidesteps these criticisms by responding that he is not “in the business of offering a strict mathematical proof for the inability of material mechanisms to generate specified complexity”. It seems he is using the word “complexity” in some sense that is all his own in a Humpty Dumpty-esque manner.
Behe’s “irreducible complexity” means “A single system which is composed of several interacting parts that contribute to the basic function, and where the removal of any one of the parts causes the system to effectively cease functioning” This requires the system to have a “function”. I don’t know if this could be said of the natural numbers.
Possibly everyone is talking about different things?
“Possibly everyone is talking about different things?”
Yes.
Agree with multiple commenters here re natural numbers. It’s as if the complexity of LEGOLAND should affirm the complexity of the LEGO block.
+1, Harold.
Harped on that very point in the previous post.
I would guess that Dawkins meant that the initial set of original rules (physical laws) that caused the big bang and set everything into motion is probably less complex than the various extremely complex systems that emerged from those (simpler) physical laws. If still in doubt, I suggest we ask Dawkins and let him defend that statement personally.
EricK:
Aren’t the Peano axioms (or any equivalent) a “simple basis for a self-bootstrapping crane which raised the integers as we know them into their present (and eternal?) existence.”?
Obviously not, for a thousand reasons. 1. Do you think there were no numbers before Peano? 2. How can the Peano axioms “raise the integers into their present existence” while still not managing to imply all their properties? There are also 3 through 1000, but I’ll stop here.
If you want to understand these issues, you might start by reading The Big Questions (the book).
The integers are simple, while the complexity of deciding what can be proved with the Peano axioms is not so simple.
Likewize, atoms are simple, while the complexity of assempling living beings from them is not so simple. What’s the problem?
Sure, there were integers before Peano, just as there were atoms before life on Earth. The Peana axioms immply the properties, but you could also say the physics of atoms imply the properties of life.
Roger: You say the integers are simple. That, of course, depends on your definition of simplicity. It is a fact that the set of true first-order statements about the integers is non-recursive. I think that’s a pretty good definition of “not at all simple”.
From the integers, you can define a non-recursive set, and you call that not simple. Now tell me how that relates to biology. Is water simple? Is the periodic table simple? Is DNA simple?
Something didn’t have to come nothing if there has always been something.
“Why is there something rather than nothing?” is specifically about existence of the physical universe. Since natural numbers don’t exist in this sense, they can’t serve as a counterexample.
Note: I am not saying Dawkins’ argument is good, since I have no idea what it is; just that, given this quote, there is no reason to believe natural numbers are a counterexample.
Aside from the question of whether Dawkins means the same thing by complexity as you do, there’s also the question of whether he means the same thing by “cause” as you do. I suspect that in the sense that he’s talking about, the natural numbers aren’t caused at all.
I appreciate SL’s argument, but refuting Richard Dawkins is almost like shooting fish in a barrel
@19
Dawkins is a blunt instrument, but for best or worse, is a necessary one.
What Dawkins mainly wanted to say (I believe): evolution is a ramp that draws complexity of living organisms upwards. Living organisms are some of the most complex systems around, hence the complexity of some bits of the Universe (or maybe the “maximum complexity”) is being increased by evolution.
I have two comments here:
-it might be useful to define complexity in some comparable way. I suspect that to compare the complexity of say the rational numbers versus the hedgehog immune system is not an easy task
-to pick apart a book, statement by statement, and attack arbitrarily chosen one may not be the most fair thing to do. A book is rarely written in a purely logical language of mathematical proofs. Often context (other parts of the book) is important and should be taken into account.
The set of true first order statements modeled by the integers is a different set from the set of integers.
So, what you are saying is that the set of true statements about all configuration states of the universe does not have a complete set of axioms.
Dawkins’ evaluation of complexity is on configuration states of the universe. The counter-example you provide is not a configuration state of the universe, but a statement about the set of all true statements about configuration states. As of yet, no known example of this set exists.
HighlyPeriodic: I do not understand your comment. Let’s start at the end (though there are several other parts I also don’t understand.) You say “no known example of this set exists”. What set are you referring to?
Let Z be the set of integers, let S_Z be the set of all true statements witnessed by the integers.
Let U be the set of all possible configurations of matter in the universe. Let S_U be the set of all true statements witnessed by U.
Dawkins: Let x be a member of U. Then x has a “simple” predecessor.
You: Not so, because no complete set of axioms exists for S_Z. Therefore, the integers are a counter-example to the above claim.
Me: Call the statement made by Dawkins s. Then s belongs to S_U, as it makes a statement about possible configurations of matter in the universe. You are claiming that S_Z is a configuration state of the universe. I claim that you do not understand the difference between a theory and a model of a theory. Here is my reduction of the scenario:
Dawkins: Every member of Z is “simple”, i.e., has a predecessor.
You: Not so, because Theory(Z) does not have a complete set of axioms.
Me: Integers and true statements are incomparable objects, and a formula phi does not witness the untruth of (forall x)(x is simple), because phi does not belong to the integers.
Hence “no known example of this set exists.” Perhaps I should have said “it is not clear that the set of sentences that are true of the universe are in fact part of the universe.”
This is all garbage anyway, as none of the original assertions are well-defined.
Highly periodic:
This is all garbage anyway, as none of the original assertions are well-defined.
So: Dawkins makes the statement that everything which is complicated arises from something that is simple. He fails to define “complicated”, “simple” and “arises from”, so we have to guess what he means. In order to construct a potential counterexample to his statement, I have to come up with something that is complicated, but did not arise from a simple precursor, and the thing I have to come up has to satisfy this condition for a broad class of possible interpretations of Dawkins’s language.
I claim that if the set of all true first-order statements about a structure X (in some appropriate formal language) is non-recursive, that’s a pretty good definition of complex — in the sense that no matter what everyday interpretation Dawkins intends of the word “complex”, anything that satisfies this definition will probably also satisfy his.
So: Dawkins says that the current state of the universe is complex, hence has must have a simple precursor. I observe that he hasn’t told us what he means by complex, but we can almost surely capture it by the difficulty of axiomatizing the set of true statements about the state of the universe. Therefore the right analogy is: No matter what you mean by complex, the natural numbers are almost surely complex by the same definition, and therefore the natural numbers — not any individual natural number, but the natural numbers, must have a simple precursor.
You write:
“No matter what you mean by complex, the natural numbers are almost surely complex by the same definition.”
Not true, please read my previous post.
I think that you are using an extremely sophisticated result, dependent on very carefully distinguishing between objects, without using the same care in your argument.
See the last line of my previous post.
Many of your “riddles” are very complex and sophisticated. I think I disagree with you on this one. I agree with one reader who stated that refuting Dawkins can be like shooting fish in a barrel. But that is not my primary point.
I think I recall you making the point that numbers are “prior” to existence in Big Questions. Or as you say above “the first cause of our Universe could be a mathematical structure that is far more complex than the Universe itself”.
I think the operative word is “could”. A lot of things “could” be. I chose to believe (yes, believe) in the priority of existence over “mathematical structures”. We have no idea why we exist.