Quadratic Gravity is Possible

Quanta magazine has article on quadratic gravity.

I mentioned this before as an obvious alternative way to quantize gravity.

People have been saying for a long time that string theory is the only hope for quantizing gravity. It is completely false. For example, Brian Greene says in a new video:

0:00 String theory is the most potent and powerful approach that we have discovered as a species for blending general relativity and quantum mechanics.

He goes on to say that string theory is still viable as long as researchers want to work on it, and that is how science works.

Goedel Proved Hilbert Wrong

I wondered whether Goedel literally proved Hilbert wrong, as is commonly claimed. I doubted it.

David Hilbert didn’t leave a single, tidy quote where he explicitly says, “I demand completeness and consistency provable from within,” in those exact words. His vision for the formalization of mathematics, known as Hilbert’s Program, emerged through his lectures, papers, and talks—particularly in the 1920s — and is pieced together from these sources. I’ll show you where these demands are clearest, based on his own words and how contemporaries understood him, without inventing anything he didn’t say.

Completeness

Hilbert’s call for a complete axiomatic system — where all true statements in a mathematical domain (like arithmetic) can be derived from the axioms—shines through in his 1900 address to the International Congress of Mathematicians in Paris. There, he posed his famous 23 problems, and the second problem implicitly reflects this goal. He asked for a proof of the consistency of arithmetic’s axioms but tied it to a broader vision of a “definite system” that settles all questions. He wrote:

“By the solution of these problems, a definitive clarification of the foundations of analysis and arithmetic will be obtained, and the system of axioms will be so complete that all remaining questions can be decided by purely logical deduction from them.”

(From Hilbert’s 1900 address, as translated in various editions, e.g., Bulletin of the AMS, 1902). The term “complete” here doesn’t mean Gödel-style completeness yet (that formal definition came later), but Hilbert’s intent was a system where no mathematical truth escapes the axioms’ reach. In his 1922 lecture “The Logical Foundations of Mathematics” (published in From Kant to Hilbert, ed. Ewald), he’s more explicit:

“The chief requirement of the theory of axioms must be that it makes it possible to decide, by means of a finite number of logical inferences, whether any given statement formulated in the language of the theory is true or false.”

This is Hilbert pushing for what we’d now call decidability — every statement provable or disprovable within the system—which implies completeness. He wanted math to be a closed, self-sufficient machine.

Consistency Provable from Within

Hilbert’s demand that consistency be provable “from within” the system is trickier—he didn’t phrase it exactly that way, but it’s how his program was framed, and Gödel’s response hinges on this interpretation. In his 1925 paper “On the Infinite” (Über das Unendliche, Mathematische Annalen), Hilbert outlines his method, later dubbed “finitism,” to secure mathematics:

“We must establish the consistency of the axioms by means of a careful description and analysis of the methods actually used in mathematics… The goal is to establish certainty by finitary means.”

He proposed proving consistency using only “finitary” reasoning — simple, concrete logic he believed was unassailable. Crucially, in his 1928 book Grundzüge der theoretischen Logik (with Wilhelm Ackermann), he posed the “consistency problem” as:

“To prove by finite methods that the axioms do not lead to a contradiction.”

Hilbert didn’t say “from within” verbatim, but his finitist approach aimed to use tools internal to the system’s basic logic, avoiding infinite or external assumptions. He saw this as a proof that the system (e.g., for arithmetic) could vouch for itself. At the 1928 International Congress, he reiterated:

“The development of mathematical science must proceed in such a way that… its freedom from contradiction is proved by methods inherent in the nature of the science itself.”

How Gödel Saw It

Gödel’s 1931 paper (On Formally Undecidable Propositions, Monatshefte für Mathematik) directly engages Hilbert’s goals. He notes that Hilbert’s Program seeks a system where “all true mathematical statements can be derived” (completeness) and where “the freedom from contradiction can be proved by finitary means” (consistency). Gödel then shows this is impossible for arithmetic, citing Hilbert’s own framework as the target.

Why “From Within”?

Hilbert’s insistence on finitary methods — logic he thought was intrinsic to math — implied a self-contained proof. Later scholars (e.g., von Neumann, in letters from the 1930s, and modern texts like Gödel’s Theorem: An Incomplete Guide by Torkel Franzén) interpret this as “within the system” because Hilbert rejected leaning on stronger, external systems, which would undermine his foundationalist aim.

The Evidence

Completeness: 1900 address, 1922 lecture—Hilbert wants all truths decidable from axioms.
Consistency from Within: 1925 paper, 1928 book/talk—proof via finitary, internal means.

Hilbert’s words don’t always use modern terms, but his intent, as read by peers and successors, aligns with these demands. Gödel’s theorems hit precisely because they showed these twin pillars — total coverage and self-proved safety — couldn’t stand together. I can’t paste a single smoking-gun quote, but the pattern across these works is what mathematicians like Gödel and historians like Constance Reid (Hilbert, 1970) point to as the program’s heart.

Okay, I guess Hilbert was wrong, but people wrongfully conclude that mathematical axiomatization does not work. In fact, mathematics can be axiomatized as ZFC, where ZFC is consistent and powerful enough to prove all the theorems of math. It just isn't strong enough to decide every question, or to prove consistency within ZFC.

It was not wrong to try to axiomatize mathematics, not wrong to demand finitary reasoning, and not wrong to want consistency. If he really wanted to prove consistency from within the system, that is not possible, and is not really desirable. Hilbert's real error was to think it is desirable, if he really thought that.

The reason is that an inconsistent system can prove anything. If a system has a contradiction then it can prove 1=2 or anything else. So it does not make any sense for a system to prove itself consistent. It is like asking someone if he is honest. A liar will say that he is honest, and you learn nothing. To assess someone's honesty, you have to ask someone else.

If it is true that Hilbert wanted a math system to prove itself consistent, then he was wrong because that is a nonsensical thing to want. That should have been obvious long before Goedel.

People hear all this and figure that our axiom systems are not strong enough to prove what we want. But actually, Harvey Friedman's grand conjecture says that all our big math theorems can be proved in much weak axiom systems than ZFC, which is what is usually used.

Explanation of Newtonian Time

Matt Farr posted a new paper on Time in Classical Physics:

Wigner (1995, 334) describes how Newton’s “most important” achievement was the division of the world into “Initial Conditions and Laws of Nature”, noting that “[b]efore Newton there was no sharp separation between the two concepts. […] After Newton’s time the sharp separation of initial conditions and laws of nature was taken for granted and rarely even mentioned.” This is the central feature of the Newtonian schema.

Some people are so locked into this view that they say that indeterminism and free will are inconceivable. When you make a choice at a restaurant menu, it has to be determined by the initial conditions, or else the laws of physics are violated. No, that is just the Newtonian schema.

For example, Sabine Hossenfelder argues:

And according to new scientist, the superdeterminist view 5:20 naturally raises the possibility that the laws of physics are at odds with unlimited free will. 5:26 What are we to make of this? For one thing, this free will assumption in quantum physics, despite 5:33 its name, has nothing to do with what we normally refer to as free will in none of the definitions 5:40 that philosophers like to use.

Regardless of what you think quantum physics exactly means, the laws 5:47 of physics are always at odds with unlimited free will. This is why they're called laws. If you jump 5:54 off a bridge, you'll fall down. And no amount of free will is going to make you fall up.

She is saying that the Newtonian schema leaves no room for free will. If your initial conditions have you jumping off a bridge, the laws of physics determine your fall, and free will cannot do anything.

I think she is alluding to philosophers who try to define free will as being compatible with all your choices being determined before you were born. To those philosophers, free will is just in your imagination, and has nothing to do with the laws of physics or any actual choices you make. Most philosophers have such a nihilist view.

Yes, the Newton schema assumes that the past determines the future. That is not a law of physics. It is just an assumption. It works well approximately in a great many cases. Not all cases, if you believe in free will.

Some people also argue that the future can determine the past, in the same way that the past determines the future.

The above paper looks at what Newton said about time, and contrasts it with relativity and Lagrangian mechanics. Everyone says Newtonian time is more intuitive than relativistic time, but I am not sure. I have no intuition for anything going faster than light, as Newtonian time allows.

Lagrangian mechanics is another story. Time is just another variable, and it is not so clear how causality works. The paper tries to make sense of it.

New Scientist just released a video:

What Is Reality? Does Quantum Physics Have The Answer?

Over the past century, quantum physics has transformed science and reshaped our understanding of reality. In this special compilation from the New Scientist archive, we trace that evolution, from the birth of quantum mechanics to today’s lab-made “mini universes.”

We explore how quantum ideas revolutionised technology, how they continue to inspire new forms of creativity, and how recent breakthroughs are pushing the limits of what we can understand.

Most of it is not too bad, but it presents an expert physicist saying, about interpretations of quantum mechanics:

I think the 5:02 one that is probably most compelling to 5:04 the majority of physicists is called the 5:06 many worlds interpretation. It's 5:08 compelling because it says that 5:09 fundamentally we are also in superposition. Every possibility has a 5:14 realization in different worlds.

No, this is crazy stuff. I hope it is not true that a majority of physicists find this nuttiness compelling.

The Schroedinger Cat was once an example of silly thinking. Now this man is compelled to believe in many-worlds because he wants to believe that he is just like a Schroeding cat.