From the top…
28/02/08 by Tyche
“This is All Epicycles! This is Nonsense!”
Antithesis, Thesis, and Synthesis of Kurt Gödel
Kurt Gödel recognized early on in his career that while the fields of mathematics and physics were expanding exponentially, so, it seemed, were their explanations and descriptions. Hidden in a footnote after just the third paragraph On Formally Undecidable Propositions of Principia Mathematica and Related Systems is Gödel’s reproach, “Definitions serve only to abridge the written text and are therefore in principle superfluous.” (38) It was with this spirit that Gödel was critical of mathematics, fond of physics, and able to elaborate on Albert Einstein’s work. The purpose of this paper is not to prove a thesis of my own devising – the subject matter itself is already glutted with proofs and theses – but rather, I intend to explore Gödel’s major contributions to science, the foundation upon which I will build my philosophical argument in an upcoming essay.
While studying at the University of Vienna, Gödel attended a lecture in Bologna delivered by David Hilbert that was to alter the course of his life. The lecture, on completeness and consistency of mathematical systems, posed the question also discussed in Principles of Theoretical Logic: “Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?” This ultimately became the topic for Gödel’s doctorate dissertation, during work on which he studied first-order predicate calculus. It was during the course of this work that he established his completeness theorem. This proved there is always a finite deduction for a logically valid formula; first-order predicate deductive systems do not require supplementary inference rules to complete proofs for all logically valid formulae. This became one of the keys to Gödel’s later work, embodied in each of his subsequent works. Take, for example, Gödel’s fifth axiom from On Formally Undecidable Propositions…:
Every formula derived from the following by type-lift (and this formula itself):
1. x2Π(x2(x1) ≡ y2 (x1)) ⊃ x2 = y2
This axiom states that a class is completely determined by its elements.
A formula c is called an immediate consequence or a and b, if a is the formula ~(b)) ⋁ (c), and an immediate consequence of a, if c is the formula
ν Π (a), where ν denotes any given variable. The class of provable formulae is defined as the smallest class of formulae which contains the axioms and is closed with respect to the relation “immediate consequence of.”24 (45)
Conveniently for us, this passage also contains a restatement of Gödel’s precept of conciseness; the footnote to the end of this passage states, “the rule of substitution becomes superfluous, since we have already dealt with all possible substitutions in the axioms themselves.” This footnote bridges Gödel’s logically necessary axiom to his philosophically necessary precept of simplicity. Gödel’s criticism of mathematics was mainly his frustration with mathematical excesses, and, after all, the entire purpose of an axiom is to define a system in which substitutions are irrelevant!
In his pursuit of simplicity and authenticity, Gödel developed one of his greatest legacies: his eponymous numbering system. This numbering system was created to simplify the extensive proofs that followed in On Formally Undecidable Propositions… and is thoroughly explained over several introductory pages in the second chapter in which its potential is immediately evident. Each basic symbol and (well-formed) formula is assigned a unique natural number. Thus, if formula C can be derived from formulae A and B via inference rule r, the entire process could be written as simply as A, B ⊢r C. Gödel described part of the process as such: “By Subst a(bv) (where a stands for a formula, v a variable, and b a sign of the same type as v) we understand the formula derived from a, when we replace v in it, wherever it is free, by b.” (43) Once established, Gödel immediately implemented his new notation.
Gödel, however, did not simply replace his formulae with their own individual nicknames; he also came to encode entire sequences of formulae, condensing entire proofs into strings of symbols. This symbol-system allowed Gödel to simplify his task on yet two more levels: with his shorthand Gödel was not only able to show parallels between statements and theorems, it allowed him to display the correlations between statements about natural numbers and statements about the provability of theorems about natural numbers.
This allowed Gödel to address that “there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms.” (On Formally Undecidable… 38) Through his codified proofs Gödel was able to clearly demonstrate that it is possible to construct an arithmetical statement that is true and yet not provable in theory, therefore any well-formed system that expresses arithmetical statements cannot be both consistent and complete. Considering inconsistency undermines logic, incompleteness is the only possible conclusion.
Furthermore, “Since, for every consistent class c, w is not c-provable, there will always be propositions which are undecidable (from c), namely w, so long as Neg (w) is not c-provable; in other words, one can replace the assumption of ω- consistency [...] by the following: The statement “c is inconsistent” is not c-provable.” (On Formally Undecidable… 70) Thus Gödel has summed up his “theorem XI” – now known as the second incompleteness theorem – stating that any effectively generated Peano arithmetic system can only state its own consistency if the system itself is inconsistent.
Many people regard Gödel’s theorems of and incompleteness as evidence that there cannot be a Theory of Everything. Scientist had been trying for centuries to link all physical phenomena, but, as we are part of the system we are describing, the universe itself is recursive and therefore cannot logically be complete. It was, in fact, a close friend of Gödel’s who was able to catalyze the search for a Theory of Everything: Albert Einstein’s theory of general relativity connected electricity and magnetism via the groundbreaking concept of spacetime. Einstein’s connection of the theory of relativity and electromagnetism gave rise to his own, less ambitious take on the Theory of Everything: the unified field theory.
By the late nineteen-fourties Gödel had become well installed in the United States where he had become a professor at Princeton’s Institute for Advanced Study. There, he was not only a close companion of Einstein’s, both personally and professionally. At this time, Einstein was struggling of the implications of his intertwining theories; he was not only left with burdens such as defining just what a quantum of light actually was, but also of developing mathematical models for his newfangled conception of the universe. Here he found Gödel to he a perfect scientific complement. Gödel succeeded in applying his synthetic mathematical principles to Einstein’s very visual abstractions of the universe.
Einstein’s theory of special relativity had established the “light cone.” In our perspective bound by Euclidean geometry, light is observed as spreading in the shape of a cone (hence the intensity ratio of 1/r2); from a temporal vantage point, space is strictly planar and therefore the “cone” observed looks simply like a circle. A light cone is this circle of light is expanding along its temporal axis. A light cone, is, in fact, a double cone it forms a double cone, the apices of which meet on the plane of space and represent our present, the cones expanding in opposite directions to encompass either the future or the past. The outside faces of these cones, referred to as “null cones,” are crucial to Gödel’s cosmological model, as they are what allow for travel into the past. Theoretically, point P at the apex of the cone may travel along its “world line” to any other point on the null cone. A “world line” is the path a body follows through spacetime, and are shown in Einstein’s field equations to be closed time-like curves. As a result of this, Gödel had mathematically demonstrated causal loops, in which not only does point P at the present precede point Q, eventually the system loops around and Q is preceding P. Thus, granted enough energy, one could theoretically travel along one’s world line until they find themselves in the past.
Works Cited
Bell, John L. “Time and Causation in Gödel’s Universe.” Publish Web Server. University of Western Ontario.
Accessed 25 Feb. 2008.
Casti, John L., and Werner DePauli. Gödel: A Life of Logic. Cambridge, Massachusetts: Perseus Publishing, 2000.
Feferman, Solomon. “The Nature and Significance of Gödel’s Incompleteness Theorems.” 2006. Institute for
Advanced Study, Princeton. Accessed 21 Feb. 2008
Gödel, Kurt. The Consistency of the Axiom of Choice and the Generalized Continuum-Hypothesis With the Axioms
of Set Theory. Princeton: Princeton University Press, 190
—. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Trans. B.
Meltzer, Ph. D. Edinburgh and London: Oliver & Boyd, 1962.
Hawking, Stephen. “Gödel and the End of Physics.” 2002. Department of Applied Mathematics and Theoretical
Physics. Accessed 20 Feb. 2008.
Hofstadter, Douglas R. Gödel, Escher, Bach: anEternal Golden Braid. New York: Basic Books Inc., 1979.
—. “Kurt Gödel.” 1999. Time Magazine . Accessed 19 Feb. 2008
Holt, Jim. “Time Bandits.” 2005. The New Yorker. Accessed 19 Feb. 2008
Okay, now some minor details that would take five minutes over coffee, but which will be very tough, given my writing style, to put concisely here. Therefore, we’ll try a double approach: I’ll give you notes here, then for further clarification (if you want further clarification) you can call me. Okay? Okay, so I’ll try to be reasonably brisk, now…
1. First sentence: it’s confusing to talk about how “the fields of mathematics and physics were expanding exponentially”, for what I hope are obvious reasons…perhaps what you’re trying to say there could be expressed better by being more specific? Possibly by saying something like: as mathematical and physical theories were getting more ambitiously integrative, Godel noticed that their actual descriptions were becoming (proportionately, do you think?) more laboured and ornate. Further to this, I have two questions for you:
(i) Was this true of physics as well as mathematics, as you say? I mean, that Godel noticed it in physics too?
(ii) Or was it possibly that mathematics was trying to accomplish its own integrative miracles, to match what was going on in physics?
You can probably skate by without fixing this. However, it might be beneficial to sharpen things up a little, lest Godel notice you…
2. Third sentence: not “it was with this spirit that Godel was critical…” but something more like (for instance) “it was in keeping with this spirit…” or “it was in this spirit that Godel conceived his criticism of mathematics…” or indeed anything at all like that, the point only being that one does not employ a “spirit” directly as a criticism, but the spirit animates or informs the criticism only. Maybe: “It was this spirit which animated both Godel’s criticism of mathematics, and his fondness for physics — ultimately leading to his elaboration of Einstein’s…” blah blah blah. These are rough notes, you understand; but something like that might serve your purpose a bit better.
3. Fourth sentence: you mean, at least, “…upon which foundation I will…”
4. Second paragraph: doctoral dissertation not “doctorate dissertation”; “during work on which” is needlessly cumbersome; and again, “bridges” says there is a relationship, but it does not describe the relationship. Take an extra twenty words and make the connection plainer, as discussed in my comment to a previous post. Then at the beginning of the next paragraph you don’t need to say “his pursuit of simplicity”, but you can say “this pursuit of simplicity”…whatever you choose to say, you can link it up better, make it more contingent on what’s in the first two paragraphs. This is flow. Sometimes you need to break flow up a little bit for dramatic purposes, to show the passage of time, to indicate that you’re going to leave off attacking on one angle and begin again on another that will eventually meet up with it, or to instantiate a brand new topic — a brief biographical sketch, for example. But just here, it’s all linked, so it should desirably look linked.
5. Fifth paragraph: do this to your first sentence — “This allowed Godel to address the fact that “there are…relatively simple problems…” Not to sound like a hipster, but see what I did there? Do that. But more than that, this paragraph needs a tad of work. I like your triumphant introduction of old Albert (well, I would!), but I’m not sure it’s just right; nor am I sure that he “catalyzed” the search for a Theory of Everything. It seems what you’re saying there is that it had always been a scientist’s dream to have a Theory of Everything, but because of what Godel has shown it’s always eluded us and always will elude us, but Einstein made it possible to conceive of achieving it. But these things aren’t in the correct order, and what you’re talking about there (if I’ve got you right) isn’t well-described as catalysis, even in metaphor. Hey, I told you I was going to be brisk! One more salient point: Einstein didn’t prove the unity of electricity and magnetism, Maxwell did — relativity deals with gravity, light, acceleration, and invariance in the (correct!) groundbreaking concept of spacetime.
Once you change this stuff, you may well see a better way to lead into the
6. Sixth Paragraph: “fourties” is of course “forties” (you knew that, I know); one struggles with things, not “of” them (you knew that again, I’m well aware); Einstein’s problem wasn’t developing mathematical models for a relativistic universe, but finding a way to reconcile the quantum mechanics birthed by his own Nobel Prize-winning theory of light with his later brainchild, relativity — and for him the key to this was, as ever, visualization: visual understanding, not mathematical modelling, was what he was looking for. But, did Godel help him find it?
Hmm, I don’t know. Einstein never came up with a viable GUT. And say, where were we again? I guess I must eagerly await your next installment so I can see just where you’re going with this…unless there is no next installment in which case what you need is a conclusion…in which case what you and I should have is a little conversation. Obviously it can’t end here! Because the crucial thing isn’t Einstein’s synthesis, but Godel’s…we know he brings his love of simplicity and authenticity to GR when he and Einstein hang out at IAS, but what’s the upshot?
No matter how well one plans, that’s always the very last unanswered question, I guess. I’m going to advise rewriting straight from “…if the system itself is inconsistent” all the way to “…Einstein’s theory of special relativity”, and then tacking on two hundred words of conclusion. And if not that, then something a lot like it.
Okay, call or be called!
Also I’m going to suggest replacing that title in terms a bit more tough: go cruise Rumi and Omar Khayyam poems on the web, and see if you can’t find something a bit more apropos. This is the tweak phase, after all — so start tweaking!
You needn’t think that the present central problem in theoretical physics is to do with constructing a theory of everything but rather one of constructing any causal theory that can be universally applied. And that the limitations in the development of any empirical universal theory are centrally to do with causation rather than logic or mathematics.
So that one basic question that needs to be asked is how much and what of the universe can actually explained given the assumption that the push or pull forces are the only causes the act universally? And so you have the problem of causally explaining the evidence of quantum physics, and thus of the natural organisation of matter in general on the scale of atoms, molecules and living organisms.
And then you can ask is the lack of any adequate causal explanation in quantum theory the fundamental reason why there has bee no predictively successful development of a quantum gravity theory?
And why is it that in twenty years of looking for the stuff experimentally, has no dark matter been directly detected, and while the modification of Newton’s laws, which doesn’t sound theoretically sensible anyway, also doesn’t work for galaxy clusters?
I’d definitely recommend Lee Smolin’s book The Trouble with Physics as giving a pretty objective insight into what the modern problems of theoretical physics really are. My only reservation being that, having given up on string theory, Smolin’s now into loop quantum gravity.
So my own conclusion is that It’s not, as Smolin suggests, the concept of time that’s the real problem but the foundations of quantum theory. Sort that problem out causally and then I say you’re really getting somewhere with a workable and properly scientific universal theory. Although you can then ask wouldn’t such a theory need to be just too radical for any physicist to contemplate?