Coincidentally, Bob Kurland sent this guest post on the day of miracles, a good follow up to yesterday’s post.
Feynman: “Do you know calculus?” Wouk: I admitted that I didn’t. Feynman: “You had better learn it…It’s the language God talks.” Herman Wouk, conversation with Richard Feynman in The Language God Talks, p.5.
In his very fine book, Is God a Mathematician?, Mario Livio gives a good history of mathematics and its foundational applications to science. He also discusses whether mathematics is a Platonic ideal or is a construction of the human mind—i.e. is mathematics “discovered” or “invented”? But he does not address the question posed in his title.
Now it goes without saying (although I will say it), that if God is omniscient, he knows everything and therefore, perforce, must know all mathematics. These propositions do not, however, require that reality is altogether mathematical, as suggested by Max Tegmark in his book, Our Mathematical Universe. If reality is altogether mathematical, then everything can be quantified, represented by numbers or properties that can put into correspondence with numbers. Is this so?
I invite the reader to suggest things that cannot be quantified by numbers. Here’s my list of a few such: self-awareness, consciousness (“Cogito, ergo sum”), moments of communion with God, The Holy Spirit, Jesus, love of another, shame, anger, pain, happiness, joy, feelings aroused by nature, feelings aroused by music, feelings aroused by intellectual discovery, the literary excellence of a poem, a short story, a novel, boredom on reading blog posts dealing with the reality of mathematics, etc.
Now psychologists might say that most, if not all of the above can be quantified: just use the simple 1-5 scale as, in satisfaction response surveys. I claim that, unlike measuring the mass of a steel ball or its radius, such a procedure would not yield a universal measurement—one person’s “2” might well be another person’s “4”. The qualia referred to in the above items are non-quantifiable, in the sense that a universally applicable measurement cannot be applied.
Let’s explore just one of the above in more detail—feelings aroused by music. In another post, God’s Gift to Man—the Transforming Power of Music, I’ve discussed the emotional and spiritual impact music has had on me, an effect which cannot be explained by mathematical relationships. The Pythagorean harmonies have no place in the dissonances of Bartok, Berlioz or even Mozart (Symphony #40, the Great G-Minor).
The inability of computation–mathematics—to emulate musical creativity is illustrated in a science-fiction story by James Blish, “A Work of Art“. In this tale “mind sculptors” of the future install a recreation of Richard Strauss in a non-musical volunteer. The volunteer thinks of himself as a resurrected Strauss, composes an opera, and then realizes it uses old musical devices and is not creative. At the concert in which the work is premiered, the volunteer knows that the resounding applause is for the mind sculptors, not for his musical work.
The eminent mathematical physicist, Roger Penrose, has said the mind is not a computer. Penrose demonstrates, using Godel’s Incompleteness Theorem and Turing’s Halting Theorem, that the human can know the truth of a mathematical theorem even when a computer can not.
In Shadows of the Mind he gives four types of belief or non-belief in the possibility of Artificial Intelligence (AI), that self-aware intelligence can be programmed by some set of algorithms:
- Consciousness is reducible to computation (the view of strong-AI proponents);
- Consciousness can be simulated by a computer, but the simulation couldn’t produce “real understanding” (John Searle’s view);
- Consciousness can’t even be simulated by computer, but nevertheless has a scientific explanation (Penrose’s own view)
- Consciousness doesn’t have a scientific explanation at all (the view of Thomas Nagel—see Mind and Cosmos)
The philosopher John Searle posits, as does Penrose, that consciousness has a scientific explanation , but that it will be an explanation in which consciousness is an “emergent” property of the brain’s biochemistry and biophysics, much as wetness can be explained by theories of surface tension for water.
A quantum computer (i.e. a scientist engaged in quantum computation), Scott Aaronson, has given an amusing and almost-convincing critique of Penrose’s thesis in one of his Physics Lectures. Some of his criticisms can be answered, particularly the one dealing with the Libet experiment, but I don’t propose to engage that discussion here. The critique relies primarily on two features: the activities of the mind are finite, not infinite; a computer which would be allowed to make mistakes would not be bound by Goedel’s Theorem.
Finally, note that Max Tegmark does not show in Our Mathematical Universe how consciousness can be explained as a mathematical phenomenon. He claims that this will be done in the future, but that seems to me very much like a scientism of the gaps.
If mathematics is to be the end-all and be-all of what is, then it seems reasonable to suppose that mathematics is complete in itself—there are no loose ends. A primitive view of Goedel’s and Turing’s theorems suggest that this is not so.
Faith, religion, beauty, love are non-mathematical and above the bounds of logic. As Pope St. John Paul II, said in Fides et Ratio:
Faith and reason are like two wings on which the human spirit rises to the contemplation of truth; and God has placed in the human heart a desire to know the truth—in a word, to know himself—so that, by knowing and loving God, men and women may also come to the fullness of truth about themselves.
So my answer to the question in the title is, God is much more than a mathematician.
An equally, or more, interesting questions is: is the average human a mathemetician? By the Merriam-Webster definition — a specialist or expert in mathematics — I’d say no. And by mathematics I mean more than estimating if the item we want to purchase will max out the credit card. For most, mathematics is a foreign language with which we struggle for a number of years until we age out of formal schooling. For a few it’s a fascinating subject that dances just beyond our willingness to pursue with diligence and determination. For a minority it’s a familiar friend.
So the movie “Pi” may have been a documentary? Not sure we should be speaking God’s language–it did not turn out well in the movie.
Pain is a good one for not being able to use numbers. When you’re in pain, medical people want you to rate it 1 to 10. I tried telling them I couldn’t but after a while, I just gave up and made up numbers. You can figure out what numbers yield the correct amount of pain meds in each situation but it takes a while. There was cute scale here: http://www.rasch.org/rmt/rmt264f.htm that I actually found much more accurate than the medical ones.
Penrose demonstrates, using Godel’s Incompleteness Theorem and Turing’s Halting Theorem, that the human can know the truth of a mathematical theorem even when a computer can not.
Briggs:
It might be apropos for God’s “proof” of Gödel?
John B()…good one! Goedel did have a proof for the existence of God…see.
http://www.stats.uwaterloo.ca/~cgsmall/ontology.html
Reminds me of a sweat-shirt I once had:
Nietzche: “God is dead”
God: “Nietzche is dead”
It offended an evangelical car salesman that I was dealing with, so I never wore it after that.
Sheri: my point about the qualia of pain was that the numbers are meaningful only on a very relative scale. My wife is much more tolerant to pain than I am, so her “8” (when she had recent hip and back problems) would be my 10++++. By the way is it a good generalization that women are more pain tolerant than men?
Bob: I thought that my example in the link and my comment about making up numbers would show that the scale is relative. Guess not.
Don’t know about women being more tolerant of pain. Some are, some are not. I have a high pain tolerance, yet my mother had next to none. Just depends on the person, I think. And the type of pain. Having surgery on one’s tongue is generally considered more painful that surgery elsewhere. Some parts of the anatomy are just very sensitive to being cut! Also, bruises and large cuts are often less painful than a paper cut. Which again, I hope, indicates I did get your point about relative pain scales.
Bob;
I realize the irony in my statement, but it wasn’t a joke.
I have a rather long treatise on God “proving” Gödel’s Theorem (or showing an application of it, anyway). Even split in two, Briggs says it’s too long to post. I can’t bring myself to make it any shorter. I told him to share it with you…?
I can’t imagine an “evangelical” offended by a joke about Nietzche. Was he trying to sell you? Or were you trying to sell him?
(if the latter, can’t imagine you approaching him in a tee-shirt in the first place; if the former, wow! what’s he doing in car sales – what does he do when he sells to someone all “tat”ted up? … Like Niietzche’s name is like Voldamort? I s’pose you couldn’t mention HIM either.
@Sheri,
Good one.
I broke my right arm just below my shoulder not quite 2 years ago. Because it was my right arm, I couldn’t drive, so I called the hospital where my primary doctor is at and asked for a reference for an ambulance service. When the paramedics arrived, they asked my how painful it was on the standard 1-10 scale. I told them that as long as I kept my arm still it was about a 6 or 7, but if I moved my arm it was an 11.5.
I don’t see how Max Tegmark’s theory that “reality is altogether mathematical” is disproved by the existence of subjectively non-quantifiable experiences or any other subjective experiences.
My personal belief is that Max Tegmark is completely correct and his theory explains many otherwise puzzling features of the universe, not least “why does it bother to exist?”
swordfishtrombone, if you don’t see why non-quantifiable things don’t disprove Tegmark’s thesis, I don’t think we can go further in explanation.
John B() …the car salesman only read the first line… I had to point out God’s rejoinder to Nietzsche….and it was a trade-in type deal.
Sheri…sorry, I didn’t see the link….but it still looks like the guy is trying to establish and absolute (if humorous) scale for pain… but one man’s “I’m dying” is another man’s (or woman’s) “well, this too will pass”.
swordfish trombone–I’ve read Tegmark’s book and I don’t agree that he explains the ultimate cause, i.e. “why does it bother to exist”.
MattS: I kept getting really nasty looks if I went outside the scale of 1-10. 🙂
Bob: I think I took the humorous scale as the guy’s way of showing just how subjective the whole thing is.
God is an antropomorphic projection.
@Bod Kurland:
I said *Subjectively* non-quantifiable.
As for your second point, I think the answer would be it “just does”, like numbers “just exist”.
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When I was in college there was a joke about Cogito, ergo sum. Rene Descartes walked into his favorite bar one afternoon and sat down. T he bartender comes over and says , you usual Mr. Descartes? Rene considers for a moment and says I think not, and instantly disappears.
Sheri,
Well, the paramedics I dealt with were kind of amused by it. EMTs probably get that kind of reaction more than doctors or the more bureaucratic health professionals.
@swordfishtrombone,
“As for your second point, I think the answer would be it “just does”, like numbers “just exist”.”
Whether about numbers or the universe, that is a hand waving non-response to a why question. No matter what the subject, that kind of “answer” explains exactly nothing.
MattS: Probably paramedics do have a better sense of humor. Glad they laughed!
swordfishtrombone, with respect to “like numbers, ‘just exist’ ”
the now-atheist at an earlier point in his career (s. hawking) wrote a book with the title “God Created the Integers”, taken from a quote by the German numerologist, Kronecker (he of the Kronecker delta symbol)
“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.”
which translates (if my scientific German is up to snuff),
“the Lord God made the integers, all else is the work of man.”
@Bob Kurland:
“now-atheist” (Stephen Hawking)
To the best of my knowledge, Stephen Hawking has never believed in God and his use of the tern “God”, like Einstein before him, is metaphorical.
@MattS:
“Whether about numbers or the universe, that is a hand waving non-response to a why question. No matter what the subject, that kind of “answer” explains exactly nothing.”
I agree but any ultimate explanation of the universe would have to end in something which “just is”, some people would say God, I would say numbers. I would also say it explains literally everything.
Perhaps the right question to ask isn’t *why* but “why not”.
Feelings can be topologically sorted. Sometimes you feel more pain, sometimes, less, most of the time no pain at all. You get numbers when you have a number of fixed pains you can compare the other pains with. But the pain system is not build for precise or exact pain measurement, so it doesn’t have these fixed pain levels.
@swordfishtrombone,
“I agree but any ultimate explanation of the universe would have to end in something which “just is””
“Just is” isn’t an explanation at all, of anything. The proper translation of “just is” offered as an explanation regardless of topic or form of question is “I don’t have a clue.”
“Perhaps the right question to ask isn’t *why* but “why not”.”
It’s not, because:
1. Why not? is usually used as an excuse, not a legitimate question.
2. If you do intend it as a legitimate question, it can not be answered without knowing the answer to “why?”
Well, Swordfishtrombone, some of my Caltech acquaintances would have it that when Hawking was there he attended the Episcopal Church regularly. And didn’t he somewhere in one of his books talk about “knowing the mind of God”? That hardly seems metaphorical, although it may not be the God of the Christian Church.
Various mistakes in this article:
The only thing you have identified as non quantifiable is consciousness, or qualia (subjective conscious experience). Everything else you have listed are variations of the same thing.
Since consciousness is unknown, it is not surprising that it can’t be quantified mathematically. All unknown things are non quantifiable. The geology of rocky extra solar worlds are unknown, hence unquantifiable. But it therefore does not follow that their geology is unquantifiable in principle. It certainly is not. So this line of reasoning is fallacious.
(There is a sloppy habit among many intellectually minded people, to attempt to draw conclusions from subject material this is completely unknown. This cannot be done as one cannot derive useful information from non information.)
“that the human can know the truth of a mathematical theorem even when a computer can not.”
This is a nonsense statement and only demonstrates that you don’t understand either Turing’s Halting Problem or Godel’s Incompleteness Theorem. What Godel is explaining, for example, is that any axiomatic system may contain logical or mathematical propositions that are true, that cannot be derived from the axioms themselves. This has zilch to do with human consciousness. If a human mind identifies such a proposition, it can only do so using mathematical reasoning anyway. You cannot derive or demonstrate a mathematical truth using intuition. You have to use maths.
There is much else to dislike in this article, but it is certainly no worse than any other armchair pop philosophical discussion out there…
Thank you for enlightening and gracious commentary Will. Perhaps you could explain in simple language, that a dolt like me can understand, why various attributes of cognition and consciousness (a distinction made by Thomas Nagel) are quantifiable, are what is required to make them quantifiable.
I will respond to your comments about Goedel’s and Turing’s theorems at more length later. I would indeed like to learn how to interpret those correctly and why my impression of what I had read in Penrose’s work, Chaitin’s book, and some other references is mistaken. Please teach me in simple terms that a dolt like me can understand
At the moment I’m engaged in correspondence with a prisoner whom I’m mentoring as a novice for Benedictine Oblate, and that takes precedence.
Bob Kurland, thanks for your article. I would be really interested to know what you think of the argument made against reductionism found in the short video “The Reductionist Delusion” on YouTube. I believe it is cogent. (Forgive the plug.)
Your post raises a question I’ve wondered about, so I’m not making an assertion on this (yet) but more asking a question in the form of some statements. The ultimate theory, “they” say, is some souped-up version of Quantum Mechanics. Every discussion I’ve seen takes measurable observables as being eigenvalues of eigenstates of hermitian operators. Key point: observables are numerical. But there are observables that are -not- numerical, such as polynomial invariants for knots, to give an example. So QM+ (or ++ or whatever) would still not be a “theory of everything.”
If you’d like my private email address, please drop a line to Matt.
Thanks for your consideration.
God bless.
A movie may be given a score (i.e., Rotten Tomatoes) which is meant to present one’s conscious impression distilled down to a number. This is not of interest. That which is qualia, the ‘feeling of consciousness awareness’ is not quantifiable as we have no understanding of what it is. To draw conclusions from our ignorance is to play the God-of-the-Gaps game which you rejected in your article, but were happy to indulge in when it suited your argument.
Regarding Godel, let me preempt your rebuttal by pointing out the obvious. If human reasoning is computational in some sense (and it must be) then it will be subject to Godel’s incompleteness theorem in exactly the same way any fixed computational system will be. Unless you assert that the human mind is infinite, which I expect you won’t. (Although even that in insufficient for reasons I won’t get into.) If you accept the existence of God as a premise, then you might wish to assert that God is not subject to the incompleteness theorem (whatever that means or might imply), but certainly everything else will be.
@Fr. John Rickert, FSSP:
” Every discussion I’ve seen takes measurable observables as being eigenvalues of eigenstates of hermitian operators. Key point: observables are numerical. But there are observables that are -not- numerical, such as polynomial invariants for knots, to give an example.”
The question was directed at Bob Kurland but allow me to chime in. This is confused, if not right-down wrong. An observable of a (classical) system is a function in the configuration space (with some technical conditions like smoothness, etc.). But an observable in QM is not a function but a unitary operator in the quantum Hilbert state space (plus some technical conditions, like being densely defined). To extract a number from an observable, that is, to compare the theory with the experiment, you look at an observable’s spectrum (which is usually larger than the set of eigenvalues). For example, in LQG, and simplifying things a bit for the sake of exposition, the quantum states can be labeled by certain combinations of knots and from thereon you can soup up a knot invariant into an observable.
Given that life requires some consistency and predictability to exist, and given that there is life in the universe, then the universe must be mathematical by definition.
@MattS:
Why is 1 smaller than 2? It just is. Ditto the universe (-:
@Bob Kurland:
I bow to your superior and fascinating inside information about Stephen Hawking’s church attendance. As an Englishman, I’m thinking that church attendance often goes with a certain social class and that might have been the case here, i.e., that it was just expected of him. In any case, I’m sure he’s not a Christian *now* and his use of the term “God” in his writing is very similar to Einstein’s. Einstein was *definitely* an atheist and was heavily criticised when he first visited the US because of this. (He said, lecturing a US Physicist!)
Swordfishtrombone, I’ll grant that Hawking’s use of “God” in “to know the mind of God” may have been largely a rhetorical device, given his later pronouncements. And, as you point out, attendance at Church is not necessarily a sign of true devotion.
Will N. I bow to your superior knowledge of mathematics and ability with proofs. I’m relying solely on authorities whose knowledge I respect–namely, Penrose and Goedel…
Here are some quotes:
“either…the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems.” (op.cit., p. 310).
Godel’s Gibbs Lecture, as published in Godel’s Unpublished Works and Lectures, p. 310
“What Godel actually tells us is that whatever rules of proof we have already laid down beforehand, if we already accept that those rules are trustworthy (i.e that they do not allow us to derive falsehoods) and are not too limited , then we are provided with a new means of access to certain mathematical truths THAT THOSE RULES ARE NOT POWERFUL ENOUGH TO DERIVE.” [emphasis added] Roger Penrose, “The Road to Reality”, p.377
“…since there are || sub i sentences which are accessible to us [humans] which are not accessible to you [artificial programmed intelligences] ” Shadows of the Mind, p.184. || sub i sentences are assertions that some specified Turing-machine action does not halt. This quote is from a hypothetical dialog between a robotic intelligence and its creator. In Chapter 3, in which this section occurs, the general argument that Penrose proposes is that the human intelligence can discern mathematical truths that are inaccessible to an algorithmic system, i.e. a computer “intelligence”. He makes this argument on the basis of Goedel’s theorem and the Turing Halting problem.
I don’t agree with your assumption that the human mind is computational. Neither does Penrose, nor do many other philosophers–see works of John Searle, for example.
And, by the way, I don’t think in your argument by analogy that geology on unknown worlds is unknown to us is apt. We can assume the cosmological principle, and that rocks found on the moon are similar in general nature to rocks found on earth, subject to different moons. You can not make the same argument about consciousness.
So, I’m sorry, but I don’t think I have anything more to learn from you. thanks for the discussion.
Feferman
Fr. Rickert….Thanks for your comment. G. Rodrigues gave a much better reply than I could. I’m not a good mathematician and am unfamiliar with topology (knots and all that) even as they might apply in QM….
I’ll ask Matt for your email. And I plan to look at the recommended video.
Shalom,
Bob