What Counts as Science? Further Berlinski Reflections on Popper and Gödel
Note: My father, David Berlinski, has been enjoying your comments in response to my posts from the Great Expectations conference. He sent me these further fatherly reflections to add to the discussion.
***
Many Ricochet readers were kind enough to post comments recently about some comments of mine. It seemed to me in reading them that they posed two questions, both of them expressions of some considerable intellectual anxiety, both of them restless and unsettled in nature.
The first: Can we say just what counts as science and what does not? When the biochemist Mike Behe observed during the Dover trial (over the teaching of intelligent design) that by his lights astrology was a science, although sure a failed science, his reprimand was swift. It came from every settled quarter except that occupied by the philosophers. The indifference to the philosophers was uneasy. In arguing that the line between what is scientific and what is not is immensely difficult to draw, were they really arguing that astrology is a science, and never mind its failures. If so, then why isn't intelligent design a science, or at least, scientific? This is a road easily imagined. Hence the uneasiness. Hence the importance of Popper's criterion of falsifiability.
And the second: The method! What does it come to, that method to which we all swear allegiance? The method of reason, rationality, evidence, right thinking, good thinking followed by good thoughts? No party to contemporary debate, from gay marriage to evolutionary psychology, for a moment proposes to forsake allegiance to the method. Given that, it is surprising that there should be so many contentious debates, evidence of a sort that while the method may be ineliminable in human psychology, it is a good deal less potent than generally adverted. Whence the very general peevish reaction among scientists, when challenged, that those posing the challenge have lapsed with respect to the method. Discussions of global warming faithfully follow this pattern, with each side blaming the other for a methodological insufficiency.
But there is in the usual comical give and take of these debates a residue of real anxiety, one very rarely voiced, a suspicion that any method, no matter how elaborated, will in the end turn on itself and like a snake, swallow its own tail. The stubborn refusal of many men and women to accept conclusions that they simply do not like is an example; and these examples by their nature can rarely be expanded into an argument. They remain at the level of a suspicion. So much for fancy arguments and see where it got you.
It is utterly astonishing that given the primitive level of discussion and debate, and given the incoherent longings and vexations that they express, that there should be a supreme example in which a method is systematically assessed by means of the method itself, and found wanting. This is the real significance of Goedel's theorem. It shows that in a very limited way that the human mind is as everyone had hoped greater than its methods.
Falsifiability
Popper's criterion of falsifiability is like some great rock: Too heavy to be moved and too large to be avoided. And Popper did have an attractive idea. If you cannot specify the conditions under which a proposition is false, you cannot claim that it is true. Not entirely new, this way of thinking. Logical positivists such as A.J. Ayer had said something similar in affirming their own criterion of verifiability. Popper was doubtful about induction; and so he was doubtful about confirmation. He saw a logical opportunity and he took it.
But to what effect? A proposition such as all men are mortal is not falsifiable, but it is obviously true. We all believe it.
Can the counter-example be evaded by insisting that it is not scientific, as a number of commentators have insisted? If one takes Popper's criterion for granted, then, of course, it follows that a proposition that fails to meet the criterion could not be scientific. If I am disposed to believe that no herrings are red, I will regard a red herring as no herring at all.
A shallow victory, I should think. The proposition that all men are mortal looks very much like the proposition that all photons have mass. Why should one be ruled out of court and not the other?
Goedel's Incompleteness Theorems
Everyone is tempted to appeal to these theorems, and no one, to this day, quite knows what they mean. Goedel's theorems have a hold on the imagination because they are about arithmetic, the numbers one, two, three and the numbers beyond. If anything is unassailable in intellectual life, it is the progression of the natural numbers and the familiar operations we perform on them. Two plus two is four.
The natural numbers are so deeply embedded in consciousness, and we take them so much for granted, that it was not until the end of the nineteenth century that they were themselves brought under axiomatic control. Geometry is otherwise. Euclid had in the third century BC created a very successful axiomatic description of plane geometry.
What Goedel demonstrated in 1931 was this: That in any axiomatic system of whole number arithmetic, it would always be possible to construct a proposition such that neither the proposition nor its negation could be derived from the axioms. A complex code is involved in constructing the proposition. Under the interpretation provided by the code, the proposition says of itself that it is not provable; under its interpretation in arithmetic, it says something about the natural numbers themselves. It has a double face. The proposition is true under both interpretations. Saying of itself that it is not provable, it could be demonstrated only if the underlying system were inconsistent. But the proposition also says something that is perfectly true about the natural numbers. It is true twice. And unprovable twice. So there are certain arithmetical propositions that lie beyond the reach of an axiomatic system.
What lends to Goedel's results their unearthly power is just the fact that Goedel demonstrated them in a perfectly rigorous mathematical way.
And there it is – something out of the reach of the axioms and, yet, quite obviously true. This is something that we can see. So far as the axioms are concerned, they cannot see this at all.
It is important to understand the real power of Goedel's theorem. The axiomatic method depends on the very old insight that one cannot prove everything. Some things must be assumed. It is this insight to which we appeal when we dismiss an argument by saying that that's just your assumption. If this is so, then why not conclude that what Goedel has shown is more of the same? What cannot be demonstrated in arithmetic must be added to arithmetic as an assumption.
This is to miss the point and miss the argument. Whatever the system that results from adding an indemonstrable proposition as an axiom, that system, too, will have something lying beyond the reach of its axioms. Goedel's theorem is not a one-shot deal. It extends upward to encompass every axiomatic system constructed to accommodate what the first one could not accommodate. It is devastating just because it sets vibrating an ominous tinkle about logical argument itself.
***
Note: I'll take the chance to remind you that my father's latest book--One, Two, Three: Absolutely Elementary Mathematics is now available. He declined to mention this out of some kind of misplaced modesty.
"How else will anyone know about it, Pop?" I asked him.
For a change, he didn't have a particularly good answer.
- Comment (50)
- · Quote
- · UnfollowFollow (3)
Comments :
Jan '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
There are some great one-liners in that video. I really need to read your father's books.
Jan '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
You're kidding with the 200 word limit here, right?
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
Hey, you're a member of Ricochet: That's why we have a Member Feed. The only limit there is the Code of Conduct.
May '10
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
What counts as furniture? Carpets? Drapes? Beds? Chairs? Science, like furniture, is not a defined term. It is an evolved term. We're really asking the question, How does the mind use labels that point to stuff in the world? I guess that when we see stuff our mind computes a "closeness" function that measures the distance between a canonical case and the case at hand. Is a piano furniture? If it is in my den, it is. If it is on stage at Carnegie Hall, it is not.
If I discover a new species of insect, that seems like science to me (It's the kind of thing scientists do.). If I discover an old sock behind the washing machine, it doesn't seem like science.
Oct '10
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
I find it fascinating that Dr. Berlinski Sr. rightly criticizes Popper's suspicion of induction by observing just how astringent such a view of "science" is, and rightly observes the far-reaching consequences of Gödel's Incompleteness Theorems' revealed limitations on deductive reasoning in any non-trivial logic, but hasn't (yet?) said anything about the Laplacian, or "logical," interpretation of Probability Theory. I wonder if I could make a trade: if I promise to read "1, 2, 3," will Dr. Berlinski Sr. be so kind as to review Probability Theory: The Logic of Science for us? :-)
BTW, I enjoyed the video montage tremendously, and it's easy to see Dr. Berlinski Sr.'s interlocutory style in Claire's writing as well.
Edited on Jun 26, 2011 at 9:35pmMay '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
The corollary to Goedel's theorems is that every system of logic has at least one inconsistency. Even a system as simple as arithmetic. Read another way in a logical system there are many paths to failure but only a single path to truth. This is why there are always bugs in software and the more complicated the software the more chances for failure. We all live with every day having to deal with constant security breaches.
More importantly, it is because of Goedel that central economic planning is a failure, legal systems become more convoluted as they grow and why creating a centrally administrated healthcare system is doomed to failure: the original set of expectations and rules are never enough and each attempt at 'fixing' the system leads to more failure.
Years ago Terry Gilliam created a film named 'Brazil'. If Goedel were alive to see the film he would have understood it immediately. It was about a future in which nothing works. I recommend it to everyone on Ricochet.
Jan '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
LOL! True, but if members see me start a post on logic or number theory, that's a guaranteed zero comment post.
The best definition of logic, post Gödel, is that Logic is the study of what your language commits you to. It is a study of language, not an ontological study of the relationships of abstract objects.
I say that Gödel's incompleteness theorem is based on the structure of language. The reason why any axiomatic system "swallows" itself is because language never allows an assertion to be self-denying. The act of assertion is itself the claim for truth. You can't make an assertion, therefore, that "evaluates" itself. Language forbids.
May '10
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
Claire Berlinski, Ed.:
A proposition such as all men are mortal is not falsifiable, but it is obviously true. We all believe it.
Can the counter-example be evaded by insisting that it is not scientific, as a number of commentators have insisted? [...]
A shallow victory, I should think. The proposition that all men are mortal looks very much like the proposition that all photons have mass. Why should one be ruled out of court and not the other?
The difference between all men are mortal and all photons have mass is that the latter is currently understood to be false. Perhaps you meant massless.
All men are mortal can, in fact, be tested scientifically. Hypothesize that it is true, select a representative sample of men, and track them over a period of time, noting whether they eventually die. When the study is complete, compute the probability that all men are mortal based on the fraction of dead men and the confidence interval.
There is the fait accompli problem, so it's not strictly falsifiable, but it can be subjected to scientific scrutiny to see whether it's rational to believe it.
Edited on Jun 27, 2011 at 12:04amOct '10
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
Actually, there's no connection to Gödel's Incompleteness Theorems there. It is, in fact, possible to develop software without bugs. There are just inadequate economic incentives to do so.
As brilliant as Terry Gilliam is, Gödel was way ahead of him: while preparing for his U.S. citizenship interview, Gödel had become convinced that the U.S. Constitution contains a logical flaw that would allow the U.S. to become a dictatorship!
Oct '10
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
C'mon, KC! You know I'd enjoy hanging out at the otherwise empty Foo Bar with you!
It's funny you say this. Russell and Whitehead agreed with you, which is why their major objective in "Principia Mathematica" (and oh, the hubris in choosing that title, that cameth before their fall!) was to disallow self-reference in formal systems. Part of Gödel's genius lay in proving that making self-reference impossible is itself impossible. G says "This statement is unprovable in PM," a "language" designed to disallow the "word" "this!"
I like to put it this way: "Information wants to be free. Computation wants to diverge."
Oct '10
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
My guess is that Dr. Berlinski Sr.'s point was precisely that the two statements are believed to have different truth values in spite of the fact that both are universally quantified over sets whose entirety of members are demonstrably not available to us.
Jun '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
To Outstripp
Wittgenstein argued that terms such as 'table' or 'chair' could not be defined by means of any clear-cut criterion but could be explained in terms of a family resemblance. An intriguing idea, but not one that has ever gone anywhere. We know what science is, I would suggest, in terms of its great examples, and these are its great theories. There are not many of them.
Jun '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
To Paul Snively
On many current views, photons have a small rest mass; there are disagreements about its magnitude. But the example that I used was meant to suggest a common logical form between certain propositions. Their truth had nothing to do with their usefulness in this regard.
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
To My Pop
If you use the button that says "Quote," you can address a member comment directly.
Love,
Claire
Jun '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
NotofBerkeley
Uhhuh -- This is not the proper was to express Goedel's second incompleteness theorem. What Goedel (and von Neuman) demonstrated was that if a system of arithmetic were consistent, it could not establish its consistency within the system itself. A consistency proof from the Great Beyond was fine, and Gentzen later provided just such a proof for arithmetic; BUT proofs of this sort inevitably demand systems stronger in expressive power than the system whose consistency is up for grabs. The consistency of elementary arithmetic is not something that anyone can demonstrate within elementary arithmetic. Goedel never specified the details of his second theorem; the work was undertaken by my Stanford colleague from the old days, Solomon Fefferman.
Jun '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
Paul Snively: I find it fascinating that Dr. Berlinski Sr. rightly criticizes Popper's suspicion of induction by observing just how astringent such a view of "science" is, and rightly observes the far-reaching consequences of Gödel's Incompleteness Theorems' revealed limitations on deductive reasoning in any non-trivial logic, but hasn't (yet?) said anything about the Laplacian, or "logical," interpretation of Probability Theory. I wonder if I could make a trade: if I promise to read "1, 2, 3," will Dr. Berlinski Sr. be so kind as to review Probability Theory: The Logic of Science for us? :-)
BTW, I enjoyed the video montage tremendously, and it's easy to see Dr. Berlinski Sr.'s interlocutory style in Claire's writing as well. · Jun 26 at 9:07pm
Edited on Jun 26 at 09:35 pm
it's a deal. I have always found probability theory very much like the bottom of the sea: Everything dark, and here and there strange forms.
Jun '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
Claire Berlinski, Ed.: To My Pop
If you use the button that says "Quote," you can address a member comment directly.
Love,
Claire · Jun 27 at 12:49am
Thanks Sweetie, but I knew it all along.
Love,
Your Pop
May '10
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
David Berlinski: To Outstripp
Wittgenstein argued ... family resemblance. An intriguing idea, but not one that has ever gone anywhere. We know what science is, I would suggest, in terms of its great examples, and these are its great theories. There are not many of them. · Jun 27 at 12:42am
Some the West Coast linguists have tried to run with that idea in terms of Roschian prototypes.
Another interesting approach is Gardenfors (I quote from Wikipedia)
[Clearly, the notion of family resemblance is calling for a notion of conceptual distance, which is closely related to the idea of graded sets, but there are problems as well.
Recently, Peter Gärdenfors (Conceptual Spaces, MIT Press 2000) has elaborated a possible partial explanation of prototype theory in terms of multi-dimensional feature spaces, where a category is defined in terms of a conceptual distance. More central members of a category are "between" the peripheral members. He postulates that most natural categories exhibit a convexity in conceptual space, in that if x and y are elements of a category, and if z is between x and y, then z is also likely to belong to the category]
Gardenfors' book is pretty interesting.
Oct '10
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
First of all, welcome, Dr. Berlinski!
It's a good thing Claire asked me to call her Claire, otherwise this could get confusing.
I felt exactly the same way about probability: too ad-hoc, no one understands randomness anyway, etc. Then I read the late Dr. Jaynes' book and realized that probability theory makes perfect sense once it's explained correctly. Since your new book apparently gives analysis the treatment A Tour of the Calculus gave calculus, I suspect and hope that you find Jaynes as striking as I do.
I'd also be interested in your thought on Abstract Stone Duality, which seems like a very exciting research program in reconceiving real analysis in constructive terms that could have important computational ramifications.
I know, in your copious free time, right?
Thanks so much for joining us, and taking the time to write with us!
Jan '11
Re: What Counts as Science? Further Berlinski Reflections on Popper and Gödel
Paul Snively: "Part of Gödel's genius lay in proving that making self-reference impossible is itself impossible." Agreed. Language can’t be “fixed” or repaired.
It’s useful to translate the notion of “system” into plain English. A system is little more than the body of assertions that you take to be true. The next question is whether the statements within that body true independently of each other, or only true when taken together, as a whole?
Philosophy has had a long conversation over what truth is. Is truth a correspondence of language to reality, or is it coherence with other statements that are already deemed true? The usual answer is that even if truth is correspondence, we can’t work with that. We normally use coherence, meaning that we test a statement to be true according to how it relates to other statements.
Gödel showed that you can’t prove the system from within the system. It’s impossible to test whether any statement is true, independent of other statements.
The problem with coherence is group-think. Very soon, statements are considered true only if they conform to “accepted” or “confirmed” truths.
Edited on Jun 27, 2011 at 6:51am