Making Finitary Deductions About Infinities

That is my 5-word definition of Mathematics. It is what distinguishes Math from every other field.

Some say that Math is the study of numbers, or the use of symbolic notation. But Music uses symbolic notation, and numbers are used by all the hard and soft sciences.

None of the empirical sciences ever encounter infinities. Cosmologists may talk about the universe having infinite extent, but there is no reason to believe that, and we cannot observe that. We only observe finite quantities.

And the sciences never make a finitary deduction either. An experiment might convince us of some fact, but it is really just evidence that makes an outcome 99% likely, or something like that. The experiment has to be refined and redone to become more and more sure of it.

Math has infinities all over the place. This is obviously true about work on limits, but it is also true about elementary statements like the Pythagorean Theorem. There are infinitely many possible right triangles, and the theorem gives a formula about all of them.

Even with all the infinities, the proofs always use a finite set of steps from a finite number of axioms. The proofs about the infinities are always strictly finitary.

Here is the Wikipedia definition of Mathematics:

Mathematics (from Ancient Greek μάθημα; máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers (arithmetic and number theory),[2] formulas and related structures (algebra),[3] shapes and the spaces in which they are contained (geometry),[2] and quantities and their changes (calculus and analysis).[4][5][6] Most mathematical activity involves the use of pure reason to discover or prove the properties of abstract objects, which consist of either abstractions from nature or — in modern mathematics — entities that are stipulated with certain properties, called axioms. A mathematical proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration.

Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used.

Here is Britannica:

mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.

Here are some dictionaries:

The abstract science of number, quantity, and space. -- Oxford

An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them. -- Wiktionary

The science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. -- Merriam-Webster

The study of the measurement, relationships, and properties of quantities and sets, using numbers and symbols. -- dictonary.com

The science that deals with the logic of shape, quantity and arrangement. -- Live science

The study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them. -- Cambridge

These are pretty good, but do not distinguish Math from science well. Yes, Math is an area of knowledge that includes numbers, but the mathematicians do proofs, with infinite numbers and finite arguments.

Sean M. Carroll Goes Woke on Sex

Sean M. Carroll has become one of the leading expositors of advanced Physics, but he has a lot of strange views that will make you skeptical of whatever he tells you.

The biggest is that he believes in the many-worlds alternative to quantum mechanics. This is a belief that anything is possible, and that nothing is more likely than anything. It is a complete rejection of all modern science.

He has his own rationalization that is mostly circular reasoning.

In his latest Ask Me Anything podcast, he says that he does not see a moral justification for parents spending money on their children's education. He says all children should get the same education.

He is married with no kids.

He is welcome to his opinions, but he does not describe the American situation accurately. In California, his home state until recently, the schools get about 50% of the state budget, and the poor districts get at least as much as the rich districts. The rich are not getting any better educational opportunities.

Some rich parents do send their kids to expensive schools, but the educational opportunities are not much different from public schools.

He has also joined the sex-deniers who say that biological sex is not binary. Biology professor Jerry Coyne is a big fan of Carroll, because of what he says in favor of determinism and against libertarian free will, but schools him on biological sex.

In reality, what they are trying to do is the reverse: adjust scientific reality so that it aligns with social justice. That is, if sex is a spectrum and not binary, then people of different genders can somehow feel that they are in harmony with biological reality. But that’s an example of the “appeal to nature.” The rights of people of different genders, including transsexual people, do not depend on the developmental biology of sex, or of any observations in nature about sex dichotomies.

I’m not going to discuss my claim that sex is binary; I’ve talked about it at length, as did Luana Maroja in her piece at Substack. I’ll just put it out there that the going biological definition of sex is that there are two sexes in vertebrates: males (who produce small mobile gametes) and females (who produce large, immobile gametes). There is no group that produces intermediate types of gametes that can unite with other gametes, so there is nothing beyond these two sexes.

Carroll is not a biologist. He is not to be confused with the somewhat more accomplished scientist, Sean B. Carroll, who really is an expert biologist.

I assume that Sean M. Carroll is smart enough to know the difference between male and female. But it appears that he is willing to recite nonsense in order to please his Leftist Woke fans.

I suggest keeping this in mind when listening to him. He sometimes gives pretty good explanations of textbook physics, but his opinions on big picture physics are dubious, and his moral and politcal opinions are garbage.

Einstein and the Equivalence Principle

New paper:

Einstein's Happiest Moment: The Equivalence Principle
Paul Worden, James Overduin

Einstein's happiest thought was his leap from the observation that a falling person feels no gravity to the realization that gravity might be equivalent to acceleration. It affects all bodies in the same way because it is a property of spacetime -- its curvature -- not a force propagating through spacetime (like electromagnetic or nuclear forces). When expressed in a way that is manifestly independent of the choice of coordinates, this idea became General Relativity. But the ground for what is now known as the "equivalence principle" was laid long before Einstein, affording a fascinating example of the growth of a scientific idea through the continuous interplay between theory and experiment.

As this article and Eikipedia explain, the equivalence principle goes back centuries. Einstein was very happy about using it in a 1907 paper, but it was not because gravity was a realization of curvature, as he did not even know what curvature was at the time.

It is my understanding that what Einstein was actually happy about was that he figured out a way to use the principle to use special realtivity to show gravitational time dilation.

Special relativity is often described as a theory about constant velocity, but back during the early days, say 1995-2010, it was widely understood to cover accelerating particles also. Poincare proposed a couple of relativistic gravity theories, but the geometry was not understood.

Einstgein figured out how to sidestep having a gravity theory, by saying that gravity was like non-gravitational acceleration. That was enough to figure out clocks in a gravitational field.

People often say that general relativity is needed for GPS navigation, but I don't think that is true. It only needs special relativity, and this trick of Einstein.

As far as I know, this idea of Einstein was his own, and not plagiarized from anyone else. Maybe that is why he was so happy about it.

Probability is Subjective

Ulrich J. Mohrhoff writes:

With Mermin, I also hold this truth to be self-evident (though it took me some time to get there), that probabilities are intrinsically subjective. ...

Mermin invokes the celebrated probabilist Bruno de Finetti, who wrote: “The abandonment of superstitious beliefs about the existence of Phlogiston, the cosmic ether, absolute space and time. . . , or Fairies and Witches, was an essential step along the road to scientific thinking. Probability too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our actual probabilistic beliefs.”

Taking the mind-independent existence of the external world for granted, de Finetti holds that there is no place for probability in such a world, any- more than there is for Phlogiston and the rest.

I agree with this, but do not deny the importance of probability.

All scientific theories are inherently probabilistic. Even classical celestial mechanics, the textbook example of the clockwork deterministic universe, was always probabilistic in practice. Observations in the sky always had errors, and predictions had uncertainty. Linear regression was invented to make probabilistic predictions about celestial orbits.

Quantum Computing Skeptic gives Lecture

Gil Kalai gave a lecture on the impossibility of quantum computers, summarized here:

My argument for the impossibility of quantum computers lies within the scope of quantum mechanics and does not deviate from its principles. In essence, the argument is based on computational complexity and its interpretation, and it is discussed in-depth in my papers which also include a discussion of general conclusions that derive from my argument and relate to quantum physics, alongside suggestions of general laws of nature that express the impossibility of quantum computation.

My argument mostly deals with understanding quantum computers on the intermediate scale (known as NISQ computers, an abbreviation of Noisy Intermediate Scale Quantum), that is, quantum computers of up to at most several hundreds of qubits. It is expected that on this scale we will be able to construct quantum codes of a quality sufficient for the construction of bigger quantum computers. It is further expected that on this scale the quantum computer will achieve computations far beyond the ability of powerful classical computers, that is, will achieve quantum computational supremacy. The Google’s Sycamore computer is an example of a noisy intermediate-scale quantum computer.

As specified later, it is my argument that NISQ computers cannot be controlled. Hence:

1. Such systems cannot demonstrate significant quantum computational advantage.
2. Such systems cannot be used for the creation of quantum error-correcting codes.
3. Such systems lead to non-stationary and even chaotic distributions.

Note that he does not say that quantum mechanics is wrong. He denies that quantum computing is a necessary consequence.

A lot of smart people and a lot of research funding say that he is wrong.

Maybe I am just a contrarian, but it seems to me that they should have been able to prove him wrong by now. They have not.

Dr. Bee Make Bad Argument for Superdeterminism

Jonte R. Hance and Sabine Hossenfelder posted another short argument for superdeterminism, without admitting that superdeterminism is their real goal.

It starts out complaining that a Physics Nature article about Bell Tests was not completely precise. The Bell Tests prove that quantum mechanics experiments are inconsistent with local hidden variable theories.

As Bell and others have pointed out, there are some subtle assumptions: that the experimenter can make free choices (no superdeterminism), that future does not cause the past (no retrocausality), and that experiments have single outcomes (no many-worlds). All of these possibilities are crazy, and no serious person would believe in them. So these are reasonable assumptions.

If their only point was that a precise statement would mention these possibilities, that would be fine. But they go further.

They say that some people believe that they have the free will to do the measurements they choose, and then "It is, in hindsight, difficult to understand how this as- sociation came about." That is, they do not understand how people could think that they have the free will choose equipment settings.

Understanding the implications is even more important now that the experimentally observed violations of Bell’s inequality have been awarded the 2022 Nobel Prize in Physics. Contrary to what is of- ten stated, these observations do not demonstrate that “spooky action at a distance” is real and nature therefore non-local.

The Nobel citation did not say that spooky action is real, or that there is anything wrong with quantum mechanics.

Rather, the observations show that if nature is local, then statistical independence must be violated. We should therefore look for independent experimental evidence that can distinguish the two different options: non-locality and statistical independence, or locality and violations of statistical independence.

No, they are wrong here. The Bell observations show that if nature is local, then the theory must be a non-classical theory like quantum mechanics, or else we have one of the crazy loopholes like superdeterminism, retrocausality, or many-worlds. Saying non-classical is essentially the same as saying no local hidden variables.

Their deceptive title is "Bell's theorem allows local theories of quantum mechanics". That is completely correct statement, as local quantum mechanics is what all the textbooks teach. But what the body of the paper says is that Bell's theorem allows local superdeterminism, and that is the opposite of quantum mechanics. There is no superdeterministic theory of quantum mechanics.

Believing is superdeterminism is essentially a rejection of all science in the last millennium. So is retrocausality and many-worlds. You can believe in it if you want, but it is quite wrong to say that it is required by locality.

Dr. Bee has started expanding her podcasts to covering science news. She does a competent job, and she is very knowledgeable about Physics. But how can you trust anyone who believes that no one has any free will to do experiments, and that every randomized trial is fake?

String Theory may Explain Consciousness

New paper:

Recent proposals in quantum gravity have suggested that unknown systems can mediate entanglement between two known quantum systems, if the mediator itself is non-classical. This approach may be applicable to the brain, where speculations about quantum operations in consciousness and cognition have a long history. ...

Our findings suggest that we may have witnessed entanglement mediated by consciousness-related brain functions. Those brain functions must then operate non-classically, which would mean that consciousness is non-classical.

Roger Penrose was widely mocked for advocating ideas like this. No one has made much progress on the problem of consciousness, and I am skeptical about this, and the next story.

Separately, I heard a rumor that a string theory prediction about holography has been confirmed in a quark-gluon plasma:

a big (not so well-kept) secret I heard the other day. Story goes that some accelerator lab (Fermi?) has been busy smashing heavy ion beams (Au nuclei?) together, creating a quark-gluon plasma. and measuring some QCD observable (say "A") of the chaos that ensues. According to a "holographic principle" (an AdS/CFT-type correspondence), A is equivalently described as some GR (or QG?) observable ("B") on the system comprised of a black hole that arises in the 5D spacetime forming the bulk (interior) of the shell on which the q-g plasma lives as a solution to the QCD equations. The Einstein equations for the evolution of the hole are solvable and B can be calculated. The lab has apparently successfully verified that the "predictions" given by the calculations of B agree with measurements of A. (Secret was leaked by Susskind in a recent talk which can be found on YouTube... my version includes a little reading between the lines and may not be completely accurate.... so I'll speculate further and guess that A is something like rate of change of temperature and B is something like rate of change in entropy, i.e., area of the event horizon. The plasma cools and the hole shrinks due to Hawking radiation?)

This is mind-blowing and, I think, of importance equal to, if not surpassing, that of the confirmation of GR by deflection of starlight during the 1919 eclipse... or of the finding of the Higgs.

I will be watching for more on this. Lenny Susskind gave some related lectures here and here.

Peter Woit has a new post trashing some related claims to testing string theory.

Scott Aaronson is claiming some new results about the complexity of the AdS/CFT correspondence. You have to skip over his previous blog post, where he describes the progressive thesis that he is aligned with:

just like at least a solid minority of Germans turned out to be totally fine with Nazism, however much they might’ve denied it beforehand, so too at least a solid minority of Americans would be fine with — if not ecstatic about — The Handmaid’s Tale made real. Indeed, they’d add, it’s only vociferous progressive activism that stands between us and that dystopia.

And if anyone were tempted to doubt this, progressives might point to the election of Donald Trump, the failed insurrection to maintain his power, and the repeal of Roe as proof enough to last for a quadrillion years.

I have never even heard of any Trump supporters who want anything like The Handmaid's Tale. Only liberals watch the show and read the bood, as far as I know. Also there was no insurrection, and no repeal. Abortion law was merely returned to the democratic process. Aaronson sounds like a parody of a left-wing lunatic. I sometimes wonder if he is serious.