Limits of mathematics in cosmology

Tom,

You have incorrectly stated my position. I do not believe that time is identical to number. I am still trying (along with many others) to get a handle on time, including your assertion that "time is identical to information."

I generally agree with Marcel LeBel that the most basic property of the physical universe is logic, in the sense that physical contradictions do not exist. He further states that "maths are based on logic. They are the metric extension of simple rules of logic." Based on physical reality and logic, I can construct logical circuits (AND, OR, NOT...) and from these construct arithmetic circuits. One of the simplest is the counter, which, as I noted, is the basis of QED and physically instantiates Peano's axioms. Unlike Kronecker, I do not count on God to supply the natural numbers (as long as a logical physical reality is available.) The numbers derive from physical logic. Before the logic existed, I do not believe numbers existed.

The addition operations that Lev references are further derivative and are simply the most useful of a possibly infinite set of logico-mathematical circuits that can be implemented. Since the natural 'temporal' characteristic of 11 is to be generated after 10 and before 12, I do not see that it makes any sense to ascribe a problem to the fact that 11 can be generated by many different logico-mathematical operations.

I am trying to understand why, when one has a physically real "natural" source of well-ordered integers that can be placed in one-to-one correspondence with time sequence, one would go to a non-ordered field looking for the same property. I'm sure this is related to your view that "...time is an n-dimensional infinitely orientable metric on a self-avoiding random walk", which I am trying to relate to "time is identical to information."

That's the fun of these fqxi conversations. They stimulate thought.

Edwin Eugene Klingman

"The use of these to accomplish 2 plus 9 or 4 plus 7 are of no consequence and have nothing to do with the nature of time. I think the point about "(temporal) information on how '11'..was formed" is non-sensical and based on a misunderstanding of the nature of number in a physical universe that supports logic."

Edwin,

I'm afraid you are not reading my posts carefully and missed the main point: numbers do not allow us to record adequately what's going on in nature, since temporal processes have formative/temporal structure that cannot be captured numerically. In general, it looks that with the development of quantum physics we are led to face the reality that the temporal structure of (physical) events is the only structure of interest.

Edwin,

That time is identical to information is a result, not an assumption.

Mathematical logic is a branch of mathematics, not the other way around.

Boolean logic doesn't capture the time dependent relations of quantum states, the "something somewhere" -- it only informs us of the "doing what" in a classical computation.

I can't understand why you think I have mischaracterized your position as a claim that time is identical to number, when you say " ... one has a physically real "natural" source of well-ordered integers that can be placed in one-to-one correspondence with time sequence ..." If you don't mean that the time sequence and number sequence are identical, I can't figure out what you do mean.

Neither Lev nor I assume temporal order, however. We get to the identity of time with information by different routes -- Lev (if I understand correctly) from a hypothesis that the time-dependent structure of events is fundamental, and I from a hypothesis that the 2-dimensional structure of the complex plane is fundamental.

Tom

Tom,

You are very well informed and bright, but you believe things that I do not believe.

First, one-to-one correspondence is not 'identical' to me. Identical means essentially, "the same as" and time and numbers are not the same as each other, to me. Nor are time and information.

The fact that Feynman was correct when he said "no one understands quantum mechanics" has not changed with his death. That no one of the five or more current interpretations of QM can explain it 'logically' does convince me that QM violates logic.

As for "Mathematical logic is a branch of mathematics, not the other way around.", it seems to me that perhaps the most significant dividing line among physicists is the one that divides those who believe that mathematics is a useful tool to map the territory of physical reality from those who believe that mathematics is the essence of the universe and can somehow create the real physical territory from the map. The latter is perhaps best represented by Tegmark's "Mathematical Universe". It is basically a Platonic view or reality.

Your statement "Mathematical logic is a branch of mathematics, not the other way around." is true as stated, but does not contradict my position (LeBel's position) that physical logic underlies math. This is based on the fact that physics does not support contradictions and that I can build physical devices that produce all of logic and all mathematics based on natural numbers. (I don't take real seriously any math that cannot be linked to numbers.) Once math arises, it is natural that the mathematician attempts to go back and 'capture' logic in the framework of math. But since I can demonstrate to you that math derives from physical reality, and you cannot demonstrate to me how you can derive physical reality from mathematics, I believe my view is correct. I don't expect this to change your view, because in the wake of Tegmark's paper I have come to believe that this is more a religious question than one of logic. Many on these blogs simply believe that math is primary and physical reality is secondary. This is perhaps attributable to the lack of significant new physics discoveries over the last half century, and the corresponding focus on more and more abstract math, but I suspect it really has more to do with psychological quirks. I no longer hold out much hope that the two sides will ever see the same universe.

I view math as a game, and have no objection to the wildest assumptions being employed as the basis of the game. I view physics as more constrained, and if temporal order and numbers can be linked in a useful manner, as I have described above, I see no need as a physicist to look for 2D alternatives.

Edwin Eugene Klingman

I guess everything comes back again and again to the confusion created in the last century (particularly with the advent of computing) whose source is the blurring of the distinction between *coding* of 'reality' and its *representation*.

There are many different means of coding information, and digital is one of them. But coding does not have to address the nature of information being coded, i.e. the meaning/semantics of this information. The reason confusion appeared in the first place has to do with the situation that so far mathematics has not dealt with the *meaningful* encoding, i.e. with representation. (Actually, I had to face this problem in the 80's.)

I suggested in my paper Representational Formalism in Which Syntax and Semantics Are Congruent: Towards the Resolution of Searle's Chinese Room Challenge that the best way to approach this confusion is to rely on the formalism in which syntax and semantics are basically the same. What it means is that if you use such formalism in physics (or any other science) your 'encoding' is meaningful, i.e. it is a 'faithful' copy of 'reality'. (By the way, I believe that there is, basically, only one such formalism.)

It goes without saying that, first of all, one has to face the central issue: What is an adequate view of 'reality'? The answers to such question will be investigated in this century, but the ETS formalism was developed to address precisely this question. I postulated that 'the reality' is an interconnected net of structured events. Again, natural numbers can be obtained/realized ("physically", to use Edwin's language) in this manner.

Mathematics, for obvious historical reasons, has adopted the numeric view of reality, but as I have discussed in several papers the reduction of *all events in nature* to a single event of the simplest structure (corresponding to the successor operation in the Peano axioms) although allow us to encode reality but in a very 'primitive' manner. For example, an event in QED has a more complex structure.

Lev,

I've looked at (and around) page 29 as you suggested. At first glance it appears not entirely unrelated to my "Automatic Theory of Physics" based on automata theory as the most promising means to understand both "the unreasonable effectiveness of mathematics in the sciences" and as the model of how physics theories are generated from observations. I'm sure there are significant differences in our approaches, but I do not at the moment have time to study your 98 dense pages.

You reference "the lack of any fundamental mathematical models whose roots are not directly related to numeric models," and appear to lament that, "during mankind's scientific history, we have dealt only with numeric models and, during the last century, with their derivatives."

If I understand you so far, your goal is to replace the role of numbers in scientific theory with a non-numerical approach. I do not see the need, nor the likelihood of success, for such an undertaking, but I wish you well in your endeavor.

In reply to your remarks above, I am not sure that it makes sense to state that a representation exists in which "syntax and semantics are the same". Meaning seems to be inherently related to contextual framing, and I'm unclear as to how one can combine the object and its context in one representation.

Do we see reality, or is our view unreal? Matthew Crawford, in "Shop Class as Soulcraft" has elegantly pointed out that it's possible to view exactly the same scene with two different interpretations; this doesn't refer to well-known optical illusions, but to the meaning of what we see. For example, while cleaning parts, "Previously... I had held one of those valves in my hand and examined it naively, but had not noticed the mushrooming. Now I saw it. Countless times since that day, a more experienced mechanic has pointed out to me something that was right in front of my face, but which I lacked the knowledge to see. ...the raw sensual data reaching my eye before and after are the same, but without the pertinent framework of meaning, the features in question are invisible."

This bears repeating: "the raw sensual data reaching my eye before and after are the same, but without the pertinent framework of meaning, the features in question are invisible."

In other words, the truth is the same for everybody, but it is not seen by everybody.

The meaning does not come from the visual data or information, which is the same before and after. It comes from the framework of meaning... the knowledge of the system, of which the piece being viewed is a part. The same truth or visual data is seen each time, but the meaning of what is seen changes, depending upon the framework in which it is understood. Meaning seems to derive from the real physical world. In his example, the meaning of the 'mushrooming' seen on the valve depended upon the design, construction, and operation of the physical motorcycle. Awareness of the visual data without the understanding of the system of which it is a part imparted no meaning.

In physics this may apply to something like 'fine tuning'. In one framework, this is understood in one sense. In another framework, one postulates a mega-verse.

That you believe that there is only one such formalism is sufficient explanation for me. It has also become clear to me over the last year or so on fqxi that almost all of us have developed our ideas and understanding over many years, and no one is really convincing anyone else to change their ideas. But we do seem to serve a worthwhile purpose in getting each other to clarify our ideas.

Thanks for your comments.

Edwin Eugene Klingman

Dear Edwin,

Please, forgive my sincerity, but with all due respect, I can't help but be amazed at how such formally trivial (i.e. not rich) computational concept as automaton can inspire one to view it as "the most promising means to understand both 'the unreasonable effectiveness of mathematics in the sciences' and as the model of how physics theories are generated from observations.

You see, I am a firm believer in the currently not so popular aesthetical principle that to respect Mother Nature means not to allow formally trivial models as the basis for understanding the universe, i.e. this is absolutely the least we can do as our 'quality control' (after all, each of us has only one life to live). The evolution of the universe has exhibited its awesome *constructive* power.

Best wishes,

--Lev

Lev,

As I indicated in my remarks, there are those who think complex math is the answer and those who think the answer has nothing to do with complex math. I am of the latter persuasion. I realize that the schools have overproduced physicists and assorted types, that must keep busy with esoterica, and math offers the best playground for these games.

It is way beyond me to conceive of how one could believe that a formalism, any formalism, can exhibit 'constructive' power (of the type exhibited by the universe), but that was what I referred to when I spoke of the religious aspects of these pursuits. Since you seem to be enjoying your pursuit, I wish you well. It will clearly carry you to retirement, with no fear of actually solving the problem.

Edwin Eugene Klingman

Edwin,

Since you are of the persuasion that "the answer has nothing to do with complex math", and you also don't believe in radically changing the basic math. language (as I'm advocating), what do you think the "answer" is about?

You must, then, believe in miracles. ;-)

Cheers,

--Lev

Lev,

Some fqxi players believe that the problem lies in math (as you seem to); others believe that the problem is in the basic concepts of physics (as I do). Few look to miracles as the answer.

I spent a little more time looking through your paper and it appears that most of the diagrams in your paper could be reformulated as diagrams in my Automatic Theory, and hence have some level of equivalence. Much of my approach was based on pattern recognition and how this can be used to 'automatically' derive theories of physics from (numerical) observational data. Although this was my 1979 dissertation, I decided to publish it when a 3 April 2009 Science paper, Vol 324, "Distilling Free-Form Natural Laws from Experimental Data" presented essentially the same approach as novel.

Having spent over four years on Albert's Molecular Biology of the Cell, 4th and 5th editions, Robert's Embryology, Epigenesis, and Evolution, and about a dozen similar texts, including Immunology texts, I understand why you are frustrated with math and would like to be able to build a 'Bubble Man' of bubble legs, bubble limbs, bubble torso, etc., but I find it difficult to take such an approach seriously. I mention these studies because it appears we have been focussed on solving essentially similar problems: of life, growth, and consciousness. While I have been very interested in biological problems, including protein folding, my real goal has been consciousness. My guess is that you exclude consciousness from the problems you deal with, since a search of your paper did not find the word.

My belief, after four decades of effort, is that consciousness is fundamental and not an artifact. This means that consciousness does not pop into existence when all of the bubbles are in place, but has a more essential existence as a field phenomenon that interacts with the physical world at all levels, from QM entanglement to biological growth and development. The mathematics necessary to describe the interaction of this field with matter are not complicated. The "internal" aspects of the field, awareness and free will, are not susceptible to mathematical analysis. Instead of positing the essential unpredictability of Nature in mysterious quantum fields, I place it in a mysterious consciousness field. You might be surprised how much physics falls out of this approach.

Edwin Eugene Klingman

Lev,

I don't think you mean to say that syntax and semantics are the same. "Congruence," the term you use elsewhere, does not imply "same." The technical difference is critical, because congruence implies recognition of characteristics between this object (or event) and that -- not sameness.

An "... interconnected net of structured events ..." where congruence of syntax (meaning) and semantics (language) has a temporal ordering effect and faithfully emulates physical reality results in zero difference between the emulation and the reality.

Because we know that not all syntactically correct sentences are meaningful, however, reducing the noise-to-communication ratio does require a richer -- and time-dependent -- representation of the "something somewhere" because classical computation does not capture it. Why, however, would not quantum computation do the job, with its richer structure of Hilbert space and superpositions?

Still trying to rule out mathematical models, on a sounder theoretical basis than I have yet seen.

Tom

Edwin,

By the way, I am a firm believer that "consciousness" is a convenient cop-out (practically in the category of miracles), since what we need is *one* formalism that explicates both physical reality and biological information processing: the latter could have emerged only if its fundamental principles were embedded in the former.

Tom,

1. I don't think you mean to say that syntax and semantics are the same. "Congruence," the term you use elsewhere, does not imply "same." The technical difference is critical, because congruence implies recognition of characteristics between this object (or event) and that -- not sameness.

Yes, of course.

2. An "... interconnected net of structured events ..." where congruence of syntax (meaning) and semantics (language) has a temporal ordering effect and faithfully emulates physical reality results in zero difference between the emulation and the reality.

Yes. By the way, this is *exactly* what Helmholtz was saying.

3. Why, however, would not quantum computation do the job, with its richer structure of Hilbert space and superpositions?

I guess the shortest answer is this. We need qualitatively different structured events (e.g. QED) and we need irreversibility, i.e. to retain formative history.

For another short answer, may I recommend Milic Capek's "The Philosophical Impact of Contemporary Physics", 1961 (actually it should be called "fundamental inadequacy/incompleteness of contemporary physics"), esp. pp. 135-140, 231-240, 361-374. Here is the most profound paragraph which starts at the very bottom of p.373 (just before that paragraph he was discussing auditory, esp. musical, perception):

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"Two successive "specious presents" are not separated by imaginary durationless instants, but *by their qualitative differences*. The term "separation" is misleading; it suggests separation in a spatial sense. We need to realize that the qualitative differences of successive moments of duration are untranslatable into spatial imagery. To differ qualitatively and to be distinct in space are two different notions." (his emphasis)

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(The inadequacy of the conventional physical concept of time, chronon, "separation" of temporal segments, etc. is discussed throughout the book.)

To me this paragraph also suggests that since such successive qualitatively different events cannot originate in 'space' they must originate elsewhere!!!

Lev,

Your reply is unsurprising. And the fact that you are a "firm believer" is confirmation of the essential religious aspects of our theories. Everyone I've dealt with on fqxi is a firm believer in their own theories. Not surprising when the theories typically represent a decade or more of pursuing a unique approach to understanding a personally pressing problem. To the casual observer a specific theory may seem full of holes, but the believer has long ago filled most of these holes satisfactorily, and the whole makes sense to him. Clearly mental models have been learned. It probably requires inventing a few new categories along the way, but the final result hangs together in the mind of the believer. And who knows, one of us might just be right.

You say "what we need is *one* formalism that explicates both physical reality and biological information processing: the latter could have emerged only if its fundamental principles were embedded in the former."

We are almost in agreement here, except that I believe we need *one* physical substance (real) as opposed to one formalism (abstract). Reality proceeds from something real, not something abstract. But that is the major dividing line I mentioned earlier. Roughly speaking, it separates the Unitarists from the Dualists. The Unitarists seem to recognize the primacy of consciousness and of physical reality, while the Dualists envision some Platonic world of relations that somehow plays the role of God in designing and governing our world. I have my theories of the types of mind that gravitate to each, but it's probably nonsense. Anyway, it's a fun game, and keeps many of us busy. As you mentioned, we (probably) only get one life.

Edwin Eugene Klingman

"We are almost in agreement here, except that I believe we need *one* physical substance (real) as opposed to one formalism (abstract). Reality proceeds from something real, not something abstract."

Dear Edwin,

Do you realize that practically every post by you contains *fundamental* contradictions, which you don't seem to notice. So as *the very last thing* I point out the latest one:

On the one hand, you believe "that consciousness is fundamental" (Aug. 28) and on the other hand, you just claimed "that I [you] believe we need *one* physical substance (real)". So how do you 'put' your 'reality' in the mind: you can't put trees, stars, and clouds in the mind, can you? Or, Is you 'reality' is not really real? Moreover, do you really believe that automata, which you like so much, are 'real'?

Thank you for you comments!

My best wishes to you,

--Lev

Lev,

A consciousness field, as I propose it is a *physical field* and physical fields distribute energy over space. According to the field equations I have proposed this field can essentially *condense* under big bang conditions into particles-- neutrinos, electrons and quarks. The theory produces both the fine structure constant, massive neutrinos (with approximately correct mass), neutrons with negative cores, and an explanation for the recent finding that the proton radius for muonic hydrogen is 4% smaller than QED has computed it to be (so much for the vaunted 15 place accuracy of QED). All of these explanations are currently missing from the Standard Model.

You opinions simply reflect your undeveloped understanding of consciousness, and have no relevance to my theory. This is the problem with most religious opinions, incomprehension leading to intolerance.

Your consensus understanding of consciousness as something occurring in the mind is conventional, but I'm surprised that you haven't at least heard of or considered alternative conceptions. I would suggest that you think of the ability of the gravity field to 'pull' on mass, but you probably believe that the gravity field is simply geometry.

Try not to take new ideas personally.

Edwin Eugene Klingman

For any reading the above thread, who are unaware that there is an alternate conception of consciousness, (alternative to the consensus of consciousness as artifact) the following fqxi is a blitzkrieg introduction to the topic:

http://www.fqxi.org/community/forum/topic/561

"Fundamental Physics of Consciousness" -- essay and extensive comments

Other more detailed info available here and at Amazon:

http://www.geneman.com/books/klingman_book_list.htm

Edwin Eugene Klingman

Tom,

I think we are getting to the heart of the problem. You make two statements that I find incredible:

1) "If numbers map 1 to 1 with time intervals, as you say, there is absolutely no difference between the number sequence and the time sequence. They are identical. Particularly, as you claim that "physical logic underlies math." Things that are not differentiable are identical in physics; e.g., the vacuum and the ether."

and

2) "Physics absolutely does support contradictions -- all the time. The most fundamental contradiction is between quantum theory and general relativity."

Perhaps I have been careless and said "physics" when I meant "physical reality". I try to maintain the distinction. I don't care that "physics" is contradictory, since it is clear from the state of today's physics that all of the current models, from the Standard Model to General Relativity are in big trouble. Lev thinks it's because of problems in math, I think it's because of problems with physical concepts.

I've remarked in several comments that the dividing line as I see it is between Unitarists and Dualists, loosely defined as those who believe in one physical substance from which the world self-evolves, and those who believe in some Platonic world of math that replaces God in governing the physical world.

Those who believe in the Platonic world of math can make statements such as yours, that if two entities map into each other, 1-to-1, they are identical. Mathematically perhaps true, physically false.

I have come to believe that those who live in the 'mental' world of math and logic have, to a serious degree, actually lost touch with the physical world. Drastic, I know, but it's the only thing that I can comprehend that explains the general view which I think you are expressing.

So when you say "Physics absolutely does support contradictions -- all the time" you should notice that I said: "the most basic property of the physical universe is logic, in the sense that physical contradictions do not exist." Either you are not paying attention to the words, or else you do not distinguish between "physics" and "physical reality". Contradictions do not exist in *physical reality*. The fact that they may exist in physics should simply be considered as proof that physics is off the mark.

I don't know whether we have a language problem or, as I suspect, a more serious perceptual problem, which I believe is reflected in both these fqxi discussions, and in the wider world of politics, where there appears to be a large percentage of humans who cannot (or do not) distinguish between abstractions and reality. Math is abstraction, physical reality is not.

I do appreciate your comments,

Edwin Eugene Klingman

Edwin,

How can one do science, while making a distinction between "physics" and "physical reality?" Reality is superfluous -- as Laplace said of a certain supernatural character, "I have no need of that hypothesis."

All scientific explanations are rendered by theory alone. There is thus no operational difference between abstraction and "reality." If there does exist some underlying reality beyond that which we can explain objectively, science won't find it.

Tom