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FQXi BLOGS
January 31, 2015

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Wigner's Key
By WILLIAM OREM • Jan. 15, 2015 @ 15:24 GMT

Here's Eugene Wigner, from "Unreasonable Effectiveness":

"The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it."

JoaoTrindale


I want to make a casual suggestion here, but one about which I am serious. What follows, in other words, is not meant to sound flippant.

Might the famous unreasonable effectiveness of mathematics—its spectacular success in quantifying, model-building and predicting future states of natural systems—be simply a matter of coincidence?

We tend to be amazed that math works again and again. Wigner compares the situation to a man with a bunch of keys finding that the first one or two he chooses always open the door. This would indeed be surprising, but the analogy can be read another way: how many ways are there to get into the house? The keys don't open windows; they don't open walls; they don't open the ceiling, or the yard, driveways or bushes or clouds. Keys fit those things that, well, fit keys. To note that one of your keys can do any task at all would be one thing. To keep being amazed that keys unlock doors is quite different.

Let's say a single key turns out to open a huge number of doors in Eugene's house—perhaps even an infinite number. That is surely an awesome thing, especially to a human mind: What a powerful key! Look at how many tasks it can handle—door upon door upon door!

But it still tells us nothing about what we aren't able to do with it (paint the house, grow the garden). Opening yet another door, excellent as that achievement may be—from the first discoveries in fluid mechanics to the latest in quantum chemistry—is, from the point of view of math's mysterious utility, essentially the same feat as it was the last time around. Look at that! Quantifiable things are still able to be quantified. Who knew?

WorldBankPhotoCollective


But even if endless discoveries are made using the abstraction of mathematical tools, we are not justified in assuming that we are, in this manner, making all possible discoveries. Something can be infinite (say, the set of all mathematical expressions that correspond, in some suitable defined way, to nature) without being all-encompassing (say, if that set we just described turns out to be a subset of another, also infinite set, "the set of all truths about nature").

Let's take a different approach. Part of logical positivist epistemology—the "logical" part—regarded mathematical statements as truths that can be known precisely because they are, ultimately, tautological. The whole intimidating edifice (Russell's phrase for Hegel) of modern mathematics is, in this view, simply a restating, or at best a following out, of axioms. Seen in this way, the complexities of any branch of mathematics could, by a sufficiently comprehensive mind, be immediately surmised from its axioms. If you understand that A is B and B is C, you already understand that A is C; you may not have stopped to draw out the steps, but when presented with the claim A = C, no new calculation was required per se. You already "got" that.

Positivism took it on the chin, and nobody much credits it any more. I am one of the few holdouts. But just grant for the moment their claim that math is an exercise in symbolic logic, and that any mathematical formalism is a tautology whose conclusions are, in a real sense, implicit in its axioms. Imagine, now, an "axiom bundle" as the sharp tip of an enormous glacier of implications. If the premise embedded in the tip is true of anything at all in the real world—any recurrent pattern, any stable quantity—then so is the glacier.

Wigner looks at the glacier and says: Wow! Look at all the things in nature that are mathematical! I'm saying: What you actually mean is that the axiom tip happened to correspond to at least one kind of natural thing.

Is that such a wonder?

FredHasselman


To return to our first metaphor, if we poke Wigner's key all over the house, eventually we may find a lock, then be amazed that we have a key and nature has a lock. If all the locks in the house are versions of the first lock, of course, we shouldn't be surprised when the key keeps working; in a sense, we keep repeating the same procedure. But it doesn't follow that we are understanding all that much about the house.

I'm not suggesting—I hasten to clarify, here, at the end—that there are *supernatural* truths we may be missing with the key of naturalism. I'm suggesting that there may be *non-mathematically accessible natural truths.* And no, I don't know what such truths would look like, though I note that some other folks harbor similar suspicions. Stephen Wolfram, in A New Kind of Science, suggests that traditional formulae are inadequate to the task of understanding nature, and that something else is needed (in his view, it's the empirical study of cellular automata). Thomas Nagel, whom I took to task here for his Mind and Cosmos, thinks consciousness itself cannot be explained using the default of materialism, which means there is at least one thing to nature that isn't reducible to mathematics (following the reductionist arrows from consciousness to neuroscience to biology to chemistry to physics to math, which is where, if you agree with Max, reality stops. By the way, Max, you owe me an email.).

I don't know how convinced I am by my own line of reasoning. (Here's a possible counterargument: Since all truths are logical—that is, even if Mother Nature throws dice, she does not act incoherently—then, if math really is symbolic logic, all truths are mathematical. QED.)

At least, though, what I've sketched out here is a logical possibility—and one that would explain that seemingly "unreasonable effectiveness" of Wigner's key as a tool for uncovering nature's secrets. It may just be a sampling error.

Torley

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A mathematical philosophy - a digital view
By JOSELLE KEHOE • Jan. 13, 2015 @ 20:34 GMT

I’ve become fascinated with Gregory Chaitin’s exploration of randomness in computing and his impulse to bring these observations to bear on physical, mathematical, and biological theories. His work inevitably addresses epistemological questions – what it means to know, to comprehend – and leads him to move (as he says in a recent paper) in the direction of “a mathematical approach to philosophical questions.” I do not have much expertise in computing (but do not assume the same about my readers) and so I won't try to clarify the formal content of his papers. However, the path Chaitin follows is from Leibniz to Hilbert to Gödel and Turing. With his development of algorithmic information theory, he has studied the expression of information in a program, and formalized an expression of randomness.

The paper to which I referred above, Conceptual Complexity and Algorithmic Information, is from this past June. It can be found on academia.edu. As is often the case, Chaitin begins with Leibniz:

"In our modern reading of Leibniz, Sections V and VI both assert that the essence of explanation is compression. An explanation has to be much simpler, more compact, than what it explains."

The idea of ‘compression’ has been used to talk about how the brain works to interpret a myriad of what one might call repeated sensory information, like the visual attributes of faces. Language, itself, has been described as cognitive compression. Chaitin reminds us of the Middle Ages’ search for the perfect language, that would give us a way to analyze the components of truth, and suggests that Hilbert’s program was a later version of that dream. And while Hilbert’s program to find a complete formal system for all of mathematics failed, Turing had an idea that has provided a different grasp of the problem. For Turing,

"there are universal languages for formalizing all possible mathematical algorithms, and algorithmic information theory tells us which are the most concise, the most expressive such languages."

Compression is happening in the search for ‘the most concise.’ Chaitin then defines conceptual complexity, which is at the center of his argument. The conceptual complexity of an object X is

"...the size in bits of the most compact program for calculating X, presupposing that we have picked as our complexity standard a particular fixed, maximally compact, concise universal programming language U. This is technically known as the algorithmic information content of the object X denoted H(X)…In medieval terms, H(X) is the minimum number of yes/no decisions that God would have to make to create X."

He employs this idea, this “new intellectual toolkit,” in a brief discussion of mathematics, physics, and evolution, modeling evolution with algorithmic mutations. He also suggests an application of one of the features of algorithmic information theory, to Giulio Tononi’s integrated information theory of consciousness. As I see it, a mathematical way of thinking brings algorithmic information theory to life, which then appears to hold the keys to a clearer view of physical, biological and digital processes.

In his discussion of consciousness Chaitin suggests an important idea – that thought reaches down to molecular activity.

"If the brain worked only at the neuronal level, for example by storing one bit per neuron, it would have roughly the capacity of a pen drive, far too low to account for human intelligence. But at the RNA/DNA molecular biology level, the total information capacity is quite immense.

In the life of a research mathematician it is frequently the case that one works fruitlessly on a problem for hours then wakes up the next morning with many new ideas. The intuitive mind has much, much greater information processing capacity than the rational mind. Indeed, it seems capable of exponential search.

We can connect the two levels postulated here by having a unique molecular “name” correspond to each neuron, for example to the proverbial “grand- mother cell.” In other words, we postulate that the unconscious “mirrors” the associations represented in the connections between neurons. Connections at the upper conscious level correspond at the lower unconscious level to enzymes that transform the molecular name of one neuron into the molecular name of another. In this way, a chemical soup can perform massive parallel searches through chains of associations, something that cannot be done at the conscious level.

When enough of the chemical name for a particular neuron forms and accumulates in the unconscious, that neuron is stimulated and fires, bringing the idea into the conscious mind.

And long-chain molecules can represent memories or sequences of words or ideas, i.e., thoughts."


This possibility is suggested in the light of a digital view of things. The paper concludes in this way:

"We now have a new fundamental substance, information, that comes together with a digital world-view.

And – most ontological of all – perhaps with the aid of these concepts we can begin again to view the world as consisting of both mind and matter. The notion of mind that perhaps begins to emerge from these musings is mathematically quantified, which is why we declared at the start that this essay pretends to take additional steps in the direction of a mathematical form of philosophy.

The eventual goal is a more precise, quantitative analysis of the concept of “mind.” Can one measure the power of a mind like one measures the power of a computer?"


Quantification as a goal can be misunderstood. To many it signifies a deterministic, controllable world. Chaitin’s idea of quantification is motivated by the exact opposite. His systems are necessarily open-ended and creative. Quantification is more the evidence of comprehension.

There is one more thing in this paper that I enjoyed reading. It comes up when he introduces the brain to his discussion of complexity. I’ll just reproduce it here without comment.

"Later in this essay, we shall attempt to analyze human intelligence and the brain. That’s also connected with complexity, because the human brain is the most complicated thing there is in biology. Indeed, our brain is presumably the goal of biological evolution, at least for those who believe that evolution has a goal. Not according to Darwin! For others, however, evolution is matter’s way of creating mind." (emphasis added)

You can view a video of Tononi talking about his information theory of consciousness, from last year's FQXi conference, here:



(first appeared on mathrising.com.)
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2014: Paradoxical Cats and the Physics Year in Review
By ZEEYA MERALI • Dec. 27, 2014 @ 18:32 GMT

Gabriela Barreto Lemos et al, Nature
It's become a bit of a tradition for quantum physicist and FQXi member Ian Durham to join us on the podcast each December to choose his favourite physics stories of the year. As always, Ian's gone for an unconventional top pick. I'd be interested to see if you can guess it before you listen -- and if you agree with him after it's been revealed.

This year (as in 2013), I'm posting three mini-editions of the podcast for your amusement over the holiday season. The first part was posted on 23 December, and I have just uploaded the second. So far, Ian's revealed the items that came in at numbers 5 and 4 on his list and 7 others that made the headlines this year, but did not quite make his cut. (We're aware that this is Ian's sneaky way of more than doubling the size of his list. Next year, we may have to run a physics "advent calendar" revealing one item on his list each day!)

I've already spoiled you on one news story that gets a mention on the podcast, with the image accompanying this post (though you'll have to listen to find out if it's on Ian's list or not, and if so where). This cute cat picture was produced by FQXi member Anton Zeilinger and colleagues and appeared in Nature (512, 409-412 (28 August 2014)) using a cunning technique that allowed them to "see" the cat without ever detecting the photons that struck it.

On the subject of "Seeing Without Looking" -- and if you want another hint about a story that made it on to Ian's list -- watch Dagomir Kaszlikowksi's top-prize winning video from our recent contest, which Ian brings up while explaining some of the best in physics in 2014:



We're also using the podcast for some housekeeping, with segments on the recent video contest and the current RFP and essay contest:

Part 1: Physics stories that didn't make Ian's top 5 (and why), Brendan and I review the video contest, and Ian's pick for the 5th best physics story of the year.

Part 2: The 4th best physics story of the year, chosen by Ian, Brendan and I run through the 2015 Large Grant RFP on "The Physics of What Happens," and Ian reveals his 3rd-placed physics story of the year.

And Part 3: Ian's top two physics stories of the year and our tips for the new FQXi essay contest, "Trick or Truth? The Mysterious Connection Between Physics and Mathematics."

Enjoy!

(Edited on 30 December 2014 to add the link to the third and final podcast.)
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Trick or Truth? — Essay Contest 2015
By BRENDAN FOSTER • Dec. 16, 2014 @ 02:02 GMT

Physics and mathematics -- It seems impossible to imagine the history of either one without the other. For gravity theory alone, we see so many examples of this -- from Newton creating calculus, to Einstein mining differential geometry.

But how close is this relationship? How deep does it go? Does physics simply wear mathematics like a costume, or is math in the blood of physical reality?

And so, Introducing our 2015 contest topic: Trick or Truth? - The Mysterious Link Between Physics and Mathematics.

Why does math seem so "unreasonably" effective in fundamental physics, especially compared to math's impact in other scientific disciplines? Or does it? How deeply does mathematics inform physics, and physics mathematics? What are the tensions between them -- the subtleties, ambiguities, hidden assumptions, or even contradictions and paradoxes at the intersection of formal mathematics and the physics of the real world?

In this essay contest, we ask all of you to probe the mysterious relationship between physics and mathematics. As always, we are giving away over $40,000 in prizes, including a top prize of $10,000. Please read the contest pages for instructions, full rules, and a lengthy list of sample questions to start your thinking.

For those of you familiar with our previous contests, let me mention a couple small but important changes to our rules.

First off, the make-up of our pool of finalists. Our finalist pool this year will consist of the familiar set of 30 top-rated entries (as rated by entrants and FQXi Members) plus auto-inducted Member entries. In addition, our Review Panel this year will have the power to add up to 10 more finalists of their choosing. This new rule means that ALL entries will be eligible for the top prizes. However, only the entries in the base set of 30 have the guarantee that the panel will read them.

Second, inspired by the smooth runnings of our first ever Video Contest (Show Me the Physics!), we are resetting the prizes. Our First Prize is still $10,000; Second Prizes are still $5,000, and Third Prizes are still $2,000. This year, only first and second prizes will receive Membership nominations. And, in place of the familiar Fourth Prizes, we will give our review panel a pool of money -- $12,000 -- to divide up as they see fit. Prizes could go to best "amateur" entry, most original presentation, deepest insight, or whatever the panel sees fit to do.

The contest is open to anyone, so please share this info with everyone. Good luck and good writing!
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Call for Research Proposals: The Physics of What Happens
By ZEEYA MERALI • Dec. 12, 2014 @ 13:46 GMT

It's good to take a philosophical attitude to life. Let's say you apply for an FQXi large grant. Maybe you get it, maybe you don't. Whatever happens, happens.

But what does *happen* from the point of view of physics? What constitutes an "event"? We're announcing our 2015 competition, with US$2 million to give out in grants, for projects that investigate "events". Full details can be found here.

The current request for proposals targets research on the Physics of "Events" both in physics and also in related fields including cosmology, astrophysics, philosophy of physics, complex systems, biophysics, computer science, and mathematics.

Everyday reality is largely made up of events: things that happen and don't "un-happen." These events separate the past from the future via the present, in which events are "happening." The world of fundamental physics, however, is quite different.

In general relativity, for instance, "event" simply refers to a space-time location, which may or may not coincide with something happening. The theory has no fundamental entity that corresponds to an event that "happens" or does not. In quantum mechanics, the measurement of some observable can constitute an "event," but this process is subject to differing theoretical and philosophical interpretations. The evolving wavefunction -- like the evolving matter and gravitational fields in GR -- has no "event" built in; they must be identified by reference to a laboratory or other macroscopic observer. Furthermore, it is unclear to what extent events in the macroscopic world should really be considered quantum measurements of the archetypal variety studied in the quantum foundations literature.

Thus, both of our fundamental theories of physics are by themselves event-free (or at best "event-lite"), and we face the question of how to draw a firm connection between these theories and the events that make up what happens in experienced reality. Drawing those connections -- and through the process, analyzing the notion of "event" as it appears in its many forms throughout physics -- is the aim of this request for proposals. A longer list of example questions is available here.

Initial proposals are due on February 15, 2015. If you have any questions, please contact mail@fqxi.org.
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