1. ## Re: The Mathematical Universe

Originally Posted by petercookintaiwan
we have the correct description for the very small, we have the correct description for the very big, and we also have the correct description of the very complex. All three pass every test every time, so we should see them working together, and not try to force them into a single description.
I see a number of problems with this. First of all we need to recognise that the "theory of the very big" is really a theory of gravity. It can be called a theory of the very big only because it generates singularities at very small scales. This is because General Relativity is only an approximation of the truth about gravity. It is probably a special case of some other theory in the same way as Newtonian gravitation is a special case of General Relativity.

In fact, none of the three theories is equipped to describe the gravitational effect at very small distances. Our descriptions of the very small and of the very complex don't include gravity, and our theory of gravity breaks down at short distances. If we wish to discover what happens inside black holes, or what happened at the start of the universe we need a new theory that joins the very small and the force of gravity.

This is an example of something that is described by my second thought which is that these areas are not distinct. We can easily imagine complex situations in which gravity acts on very small scales. This idea leads to the main motivation for the search of a theory of everything: if we have distinct theories, where does one cease to apply and another take over? And perhaps more importantly, why? It is certainly possible to imagine a universe where the laws of nature are not the same everywhere, but one in which the laws change not based on location, but on the number, size and separation of entities involved in a particular interaction seems awfully bizarre.

Mathematics is not a description of reality. Mathematics involves setting rules and seeing where they lead. The rules do not have to have anything to do with reality all. Often, and in mainstream mathematics, they are based on "truths" that we observe. But that does not mean that they do actually represent that reality. We started by study Euclidean geometry based on the idea that space is flat, but it turns out that it isn't. We never have (and never will) prove the existence of a quantity $\pi$ partly because we are limited in the accuracy of our measurements, but more importantly because we don't know whether time and space are discrete or continuous. If they are discrete, the quanta are sufficiently small that the continuous model of the real numbers is (thus far) are reasonable approximation, but that doesn't mean that it is exact.

The idea that we could ever work out deterministic equations for the universe seems inherently unlikely to me. Quantum theory is probabilistic, not deterministic. And even if it weren't, we know that we can't measure both the position and velocity of any particle - and we must be talking about particles at the start of the universe because of the small size of the universe at that point. We know already that there have been a lot of lucky breaks in the development of life on this planet. There are countless extinction events that could have extinguished it, and even some events that have happened only once in the history of the planet without which advanced life forms could not have formed. But even if we didn't, the apparent scarcity of life we have observed in the universe would tell us that. However, more or less every stage of the development of life has some widely accepted theory attached to it.

In The Blind Watchmaker Richard Dawkins talks about how the theory of evolution can be applied to non-living scenarios to account for the local accumulation of chemicals that would eventually combine to form life. He mentions lab experiments in which the building blocks of life have "spontaneously" appeared in test tubes of suitable organic compounds. In essence, there are reasons to believe both sides of the argument: that life (of some sort) is inevitable given the appropriate "initial conditions" that eventually manifested themselves on earth; but also that those initial conditions were rather a fluke and that subsequent events might easily have terminated the development of life.

I'm not an expert on Complexity theory at all, but my reading of chaos theory is not that it tells us how things happen. Instead, it relates how, although tiny variations in initial conditions might create huge differences in later results, there are still overarching principals that apply whatever those initial conditions may have been. These principals are seen more as describing results than causes. I would expect that such results have causes that are described by other branches of science, notably physics.

Your claim that "physics is set and unchanging but nature is flexible so we need a description that reflects this" is somewhat baffling to me. Physical theories change all the time as the result of research. There is also some possibility that some of the laws of nature have also changed (although not since the very early universe). But even if you refer to the "unchanging" laws of physic that science is trying to uncover, it still makes little sense. We have very, very simple games (cellular automata) such as Conways Game of Life that prove that extremely complex results can emerge from even the simplest unchanging rules. Why the laws of physics should be any different, I don't understand.

2. ## Re: The Mathematical Universe

Thank you Archie, that is an excellent reply.

I take your point about simple games leading to very complex results, but the gap between the staggering degree of complexity inside the simplest bacteria and the laws of physics is huge. I have not read anything that comes close to bridging that gap, so I am suggesting a different approach. A cell is full of interdependent systems which build into incredible complexity, but quantum physics cannot even get started with our present understanding. You can hear trusted scientists on the BBC program In Our Time, The Second law of Thermodynamics, saying anything which self assembles is negative entropy. Simon Conway Morris calls this his favorite description of life. Isn't negative entropy positive self organization? It is just another way of looking at it. The same program ends poorly when Melvyn Bragg the presenter questions the traditional approach and says , (quote). Doesn't it give you give you any cause for thought that this (life), is contradicting the Second Law of Thermodynamics? Things are not getting worse, things are not wearing out, things are not becoming more disordered, things are becoming more complicated.

You say we are driven into complexity by greater disorder elsewhere, but where is this elsewhere.

Scientist, "On the Sun". it is the dispersal of energy on the Sun that 'you think' is giving rise to complexity.

Melvyn Bragg, "There does seem to be an element of getting away with it".

This program is still on the science archive for you to listen to. I am sure most people will find this answer unacceptable. Life is built upon chemical systems which cannot have anything to do with what is happening on the Sun. Complexity theory on the other hand can show self organization building up in systems, but complexity theory is pure mathematics and does not need atomic forces to organize the system. So we already have a picture on the table that allows chemistry and life to be built by a combination of atomic forces and complexity theory. We already have on the table a picture that allows the Solar System to be built from a combination of gravity and complexity theory. But the physics of forces is in separate books to complexity theory so we do not get a consistent picture.

There are some great charts used in Asia which show reality as the combination of fundamental forces and complexity theory, and students find them easy to deal with. Complexity theory runs all the way from chaotic to self organized criticality, so it has the ingredients missing from physics to give us the order produced in complex systems. I will quote Robert C Hilborn with his take on the problem. "First and most importantly they are systems that are driven away from thermal equilibrium. As scientists we would like to explain the emergence of order on the basis of known forces".

There is still a long way to go, but the perspective I see in Asia may be a start. Thanks again for the very well written reply.

If you want a copy of the charts, please just e mail hydrogentohumanproject32@yahoo.com

3. ## Re: The Mathematical Universe

Another thing needs to be mentioned given your response post.

The mathematics of high dimensional spaces is very young to say the least - and this includes not only geometry but also the algebraic approach. Geometry concerns itself with distance and angle primarily but the algebraic approach is a bit more subtle and within linear spaces we look at linear algebra. The connections are definitely there between algebra and geometry but they can be subtle.

The point of this is to point out that analysis of high dimensional systems is not that well formed. Subjects like operator algebras and things born from the work of people like Von Neumann and Hilbert are young and there is a lot of research being done on them. I would imagine it is going to take at least another hundred years to get it to the point that the single dimensional frameworks are at now.

If science ever has a hope to make sense of and describe the world then the multi-dimensional analytical framework needs a lot of maturation.

Also what you are talking about is a problem of uncertainty and information and if you are to model a system with certainty then you need access to the information to do so in addition to being able to form the relations to make the description of said system.

If you don't have the information then it is highly unlikely that you will get the rules. Science has two postulates - redundancy in process and redundancy in information. In other words we assume that the dynamics of the universe act the same across the geometry of the observable (and hopefully unob-servable) universe and that the structures in the universe are also the same (again across the geometry of the universe). So far, this assumption has held pretty well and science along with its applications have used this to great success.

However even with redundancy the ability to access information and confirm redundancy (and reduce variation) is hard and when non-local theories of information exist then it makes things even harder. If you can't access the information then you can't reduce the necessary uncertainty and an example of this in theory and practice is the uncertainty principle relating velocity and position. Note in statistical inference the theorem stating uncertainty as a relation to the density of information in a sample is well established.

So there are two real problems here - the first is the lack of a mature multidimensional mathematical framework and the second is the lack of information - or at the very least the lack of putting existing information together in a way that optimally reduces uncertainty about the description of a known system that is represented in said information - like the observable universe.

If you really want to answer the question you have you need to be able to estimate the uncertainty in some way and if you can understand the information and its dynamics then you could create some theorems to make progress in doing so.

Understand that modern inference is not deterministic - it is based on randomness. The thinking of the deterministic logic has long passed and we are in the age of statistics and probability - not of deterministic logic and predication.

4. ## Re: The Mathematical Universe

Thanks Chiro, I think you really nail it in the last couple of sentences. Correct me if I am wrong, but people have previously thought that randomness would lead to more randomness. Here are some findings from a research group in UT Dallas. "We have studied how random fluctuations inherent in all biological systems can lead to ordered behavior in those systems. We discovered that rapid fluctuations play a crucial role in this behavior, although it has been ignored in almost all previous studies".

I take your point about reducing uncertainty in a description of the universe, but if we take any step in the right direction it must be better. Traditional physics still works perfectly well at a very simple level, two bodies in space or a lever, so I think we need models that allow for a combination of not only randomness, probabilistic and deterministic behavior, but also the incredible amount of self organization we see in the Solar System and life. The details are going to be endless but a combination of physics and nonlinear mathematics does contain an explanation for all these aspects of the nature we observe. This view point is not expressed to university students who say they feel confused by the variety of answers people give. You last sentence would be very thought provoking to many professors of the older generation. Thanks again, great ideas.

5. ## Re: The Mathematical Universe

Self-organization is probably best represented through low entropy. Just be aware though like entropy has a basis - just like a geometry, set of orthogonal functions, set or other structure.

In terms of how information works with statistics, we typically try and derive estimators that maximize the information content of a sample and produce a consistent unbiased estimate of some parameter. This is what all of those confidence intervals are all about and provided the assumptions are closely met it provides a way to get better evaluations as the information density of a sample increases. Evaluating assumptions however is difficult. Statistical inference usually starts with an assumption of complete uncertainty relative to a distribution but if you want the worst case scenario in terms of entropy, it is always going to be a discrete distribution that is uniform where the conditional distribution given two events is always classed by independence where P(A = a,B = b) = P(A=a)*P(B=b). In terms of information, you will have the most amount of uncertainty and require the most amount of information to make a resolution and to estimate a parameter with some given level of uncertainty.

In terms of what mathematics does in things like science - it can provide the invariants that characterize particular classes of systems and representation. As an example the determinant is used to classify whether a system is invert-able or not and this one invariant does that for all general linear operators. Similarly, other invariants in different fields of mathematics can do the same thing. Mathematics allows us to look for invariants that may put other invariants like physical invariants into greater context and the benefit from this is that the mathematical ones need no prior understanding about the system - what is often needed is to have a sense of what one is looking for and then attempting to clearly define and mathematically evaluate (or measure) it. Note that dimension is also a good invariant that can help with estimating complexity - and note that it has a non-linear version as well.

The thing with probability and statistics though isn't so much about estimation but more so with representing relationships statistically that don't change. With financial engineering there is a focus on stochastic analysis and stochastic calculus where we represent random variables and their measures instead of deterministic variables. What this means is that you need to resort to measure theory and results about probability theory to be able to integrate random variables to get the analogues in deterministic calculus.

Because of this current set-back one doesn't quite yet have the tools to do similar kinds of modelling and fitting in the way that is done in normal deterministic applied mathematics. This is much like the multi-dimensional mathematics mentioned above where that is not yet fully matured either in the way that single variable and deterministic mathematics is.

One other thing that is important to point out is that probabilities are conditional (most anyway) and although this may be obvious to scientists and statisticians - it's not always obvious to those using results. Bayesian methods are very new and I am aware that a lot of probability fields that involve conditional distributions and measures (like covariance, correlation, and similar) are actually extremely new with some techniques being considered very cutting edge. Conditional distributions just imply that context exists and that absoluteness does not. You should be aware that in terms of information density, correlation and co-variance have a big impact on this in addition to measures based on conditional distributions.

The importance of getting some of the stochastic calculus right for use in science is based on the principle needed to relate any two quantities together - that of change of the derivative. You may be able to fit a mean response model based on two variables and that is easy to do with regression modelling. However being able to do it with much more general forms of random variables and measures is difficult - but necessary to be able to understand how two random variables can change with each other. We know that if the inverse function theorem holds across a wide interval then we consider two things independent in a deterministic manner. Probabilistic-ally speaking, we use separation of probabilities (mentioned above) to establish independence. If things are independent then it adds to the dimension and information density of a system.

In terms of randomness leading to more randomness that depends on the dynamics itself. You also need to define the basis just like you do with say a vector space, inner product space, orthogonal function space, wavelet space, or something else. You can represent one thing in many ways and in one basis you get very little information content while in another you get quite a lot. The challenge of mathematics, science and engineering is to find a suitable basis and make sense of why it is suitable relative to all the other bases out there.

A self-ordering system is going to create relations between different information points and this means that dimension and information density is reduced. It's like if you start off with say 10 independent information points and if relations are created between pairs that are able to define how they change over time and given current value then it means that this now reduces to 5. If this happens even more then we reduce dimension and thus information density even more. If relations do not form then independence - and thus information density is kept the same. Again the inverse function theorem and separable joint distribution provides a way to define this condition.

You can also combine the above with statistics and estimation.

6. ## Re: The Mathematical Universe

Goodness me, I am going to need some time to digest this one. I really appreciate people who understand things at this level giving me time. I know how many years it takes to take this stuff on, terrific stuff!

7. ## Re: The Mathematical Universe

Hello Peter,

Hilborn is a wide ranging book and would be on my reading list, however I think that like many his definition of Nonlinear is too narrow or too specific.
Most of the phenomena in the known universe are nonlinear in nature, but we do our best to 'linearize' the mathematics to make it tractable.

I observed before there are areas of mathematics with no known real world embodyment; likewise there are phenomena in Science that defy current mathematical definitions, such as the hydraulic jump.

Final point, I would welcome your definition of the term 'random' since having an adequate definitions underlies much of modern mathematics and science.
For instance consider the set {0,1}
What is a random number in this system?

8. ## Re: The Mathematical Universe

Originally Posted by petercookintaiwan
the gap between the staggering degree of complexity inside the simplest bacteria and the laws of physics is huge. I have not read anything that comes close to bridging that gap
Why would you expect particle physics to tell you anything about biological or chemical systems? Do you expect it to tell you about traffic flow in city centres? It seems to me that you are confusing the mechanisms with the processes. We use chemistry and particle physics to describe events that might happen inside stars, but then use statistical inference and presumably complexity theory to determine whether it is reasonable to suppose that such small-scale events would produce the large-scale results that we see in the universe. This is not a new approach, but neither does it suggest that physics, chemistry or biology is particularly flawed.

I'm very amused by your citation of "In Our Time". In my experience it is equivalent to the books "A Dummies Guide to...", but with rather more inaccuracy and fudging due to time restraints and a drunk host.

9. ## Re: The Mathematical Universe

Yes I agree that Hilborn's book is narrow, but in fairness to him it was written in a time when many of the ideas expressed in this forum would have been unacceptable to most. It is a book with many daring ideas and he is not shy of predicting that nonlinear dynamics will reshape the future of humanity. I quote, "The number 4.669 . . is destined, we believe, to take its place along side the fine structure constant in the pantheon of universal numbers in physics". So he is already saying that we must see reality as a combination of physics and the constants that underpin nonlinear dynamics. You have the ability to see things from a very broad view point, but many professors still do not and teach tradition rather than what we understand.

Thank you for the challenge to define random. This is just the sort of issue which needs to be addressed in a modern world. I work with a discussion group of academics and industries best thinkers and we worked late last night on this one. We have tried to come up with an answer that can be understood by anybody, so please forgive us if it does not contain technical terms. We have to present argument to people who will not fund us if they cannot understand what we are saying. We would also like to say that we believe that the words we still use are the same words that described very different ideas in history. We don't change the words but maybe we need to think about it more.

We would argue that a single entity or number cannot be described as random, random means we have to consider relationships as random because something is only random when considered relative to something else. If we consider the movement of chunks of ice in the rings of Saturn, we would argue that each piece has complete freedom of movement, but it will behave according to the rules that govern classical objects, so only it's path is random, it still has to obey the rules. You and many like you are willing to accept that the universe is like a game of chess, we have set rules and parts that allow unpredictable play. These are much more recent ideas than many people think, there are still many reductionist s who would not agree. Dare we add the argument of free will, you are completely free to behave and think as you want but you will do so within the boundaries of human behavior. This is like saying that the chaos inside the rings of Saturn is tidied up very nicely into a beautiful mathematical structure. The disorder in the Solar System sits where it should in the rings and belts. We are not overlooking individual objects that roam outside.

There are always two answers which have to be considered at the same time, so if we look at quantum numbers, when we say the charge is 1, we mean one unit of discreteness but not the actual unit of electrical charge which is a fraction of a coulomb. Many mathematicians are reluctant to consider dual answers but if we take the number 7, we must consider it as being able to behave as a single unit of 7 or a set of individuals each behaving as 1. I know that a solution to Riemann's Hypothesis has been put forward that reflects this but I don't know the details or the outcome, it just means some people are thinking this way.

We came to the conclusion that something can be random relative to an aspect of something else but it will also have boundaries that are not random, a mix of order and disorder if you like. The ratio of the mix will have a mathematical value that governs the system.

We are not arrogant enough to say this is true, it is just being put on the table for consideration. We can back up all we say giving good example in each case but we wanted to write it as simple as possible. The simple example we use considers the randomness of getting a male or female leading to en even distribution within the population. We cannot consider just the single event, but we must consider the individual event and the collection of events at the same time.

I will put this view point to physics PhD students next week. Kind Regards P A Cook

10. ## Re: The Mathematical Universe

I am not arguing that particle physics will tell us anything about traffic flow, I am arguing against the people who think that one day it will. Let me quote Roger Penrose, " The major revolution I do think we need would involve a complete rethink of not only quantum mechanics, but how we look at space and time, so there will be a revolution waiting in the wings and when that revolution has come maybe we can think about the issue of what conscious thing is, and my view is conscious thinking does depend on the unknown part of physics". So you can see the problem, Roger Penrose saying that an answer to what thought is will come out of physics.

We know that the mathematics of traffic flow or any other complex system is all related and not just ad hoc, but we are asking Roger Penrose to consider a model of reality that does not just depend on physics, but a combination of physics and nonlinear mathematics. We would like to argue that nonlinear mathematics does not have to one day turn into physics but can remain as pure mathematics even though it governs the behavior of physical parts.

The perspective we would like to put forward is that we see physics emerge out of the pure mathematics that describe the base entities and the start of the universe, not the other way around. The description of these entities as numbers only is covered by Max Tegmark in his book The mathematical Universe, p164/165 and it is hard to argue with. Beneath physics we only have mathematical descriptions but we do not see the Standard Model being talked about in books on number theory, that's not true, they are being talked about but it is seen as a coincidence, and I have had PhD mathematics doctors from Harvard say that straight to my face.

That is why I wanted to start this discussion, Roger Penrose has very high social status which means people listen and trust. There are many others in education and industry who still think like this, but the people who have given answers in this forum have shown that many are ready for a change. I must say that the people I work with in the universities in Asia have been staggered at the quality of both questions and answers, so thank you for your contributions, they have produced much respect.

I can direct you to the Roger Penrose quote if you like.

11. ## Re: The Mathematical Universe

The mathematical universe is not the real universe. Mathematics doesn't govern anything, but it does provide models that can be used to approximate the real world.

It is likely that the future will lead us to greater insights into how the brain works, but whether we will ever be able reliably to read or even predict thoughts is another matter. That is probably something that would require too many initial conditions to be read, many of them at too low a level to be accurately measured. We are likely to have to make do with broad brush probabilistic theories that described an "average" expected response to basic stimuli. Complexity theory may have something to say about this, but it won't be able to give definitive answers because those initial conditions will remain unknown.

12. ## Re: The Mathematical Universe

I'm not sure what the actual thrust of this thread is.

Our scientific model is "empirical", we observe, conjecture, test our conjectures, and repeat. By all accounts, it is an imperfect system, and will likely *always* be. For one, there are limits to what we can physically investigate. For another, we can only conceive of things we already have. The underlying "truth" (if there is, in fact, an objective reality) may be something totally incomprehensible to us, or as yet alien. We may also never know. That said, we measure our science by the reliability and utility of the results it has, and the predictions it makes. It may not be perfect, but it's our "best shot".

Mathematics, and science, have an uneasy partnership. For my money, mathematics is the language of pure thought-which sometimes lends itself to expression of other realms (such as physics, or accounting). The ideas in mathematics are not constrained to model reality, but it is perfectly acceptable to use our real experiences in creating mathematics. As Goedel showed, mathematics has its limits, too. I believe this is intrinsic-I see the universe as being like that elephant the blind men thought was various things: we will likely *never* have "the whole picture", but we may have reasonably detailed information on "part of it".

I am leery of ideas having credence "just because they're Asian." While some Oriental philosophies have some unique insights, and may be superior in some respect to "classical Western rationalist thought", I see no real difference between the human condition in Europe or the Americas versus Asia. Everyone still fights to survive, and eventually loses the struggle-there are the same battles with hunger, poverty, the oppression of the weak by the strong, political in-fighting, disease and over-population. As I see it, not one nation on the Earth has the foresight to fight against our species' eventual extinction. So for all of humanity's accomplishments, we're just "spinning our wheels". We went to the moon, and decided: "that's far enough" A poor showing, in my opinion. In the meantime, we are poisoning the planet we live on, and pretending it isn't happening.

My point being: Physics? Mathematics? What difference does it make? We might as well discuss who is the best boy band of all time.

I do not mean to disparage any one person's ideas-some of the posts here are quite thought-provoking. It's possible that one of the intellectual giants, such as Penrose, or Tegmark, or one of their intellectual heirs, will solve some great mystery, perhaps something worthy of being called truly transcendental. Not holding my breath.

13. ## Re: The Mathematical Universe

Randomness is best described in terms of entropy and as mentioned above - entropy has a basis.

As a function of information, if something is random then no information in any basis is going to give any more predictive power and from a point of probability this means probabilistic independence.

Randomness is a property of information - no matter how you look at it whether it be using Kolmogorov complexity, entropy, or some other measure. If information does not increase the ability to reduce uncertainty about the dynamics and values of a system then it is an indicator of randomness.

We also have to distinguish the inability of a person (or people) to lack identifying information and relationships versus them actually existing. Just because people (even educated and intelligent) can't find the relationships and its basis does not mean it doesn't necessarily exist.

A final thing I should mention - there are many ways to organize information - including within the framework of mathematics and the ability to organize information yields its understanding of it. If information - even in mathematics is not organized well enough to ascertain the relationships that are sought after then it is unlikely they will be sought unless information is re-organized. The organization of information yields the understanding of it and one should always examine whether they can reorganize information to obtain better relationships between pieces of information and in the process gaining new understanding.

There are many ways to organize information and each form can yield something another may not.

14. ## Re: The Mathematical Universe

Originally Posted by petercookintaiwan
the gap between the staggering degree of complexity inside the simplest bacteria and the laws of physics is huge. I have not read anything that comes close to bridging that gap
In a standard undergraduate course on quantum mechanics, one learns the basics of aufbau. From that basis one learns the fundamentals of chemistry. The mechanisms of organic chemistry are well-known, and again are taught at undergraduate level. The chemistry of DNA and the genetic code was solved by the middle of the last century (I even have a book written by Isaac Asimov, called "The Genetic Code", which explains it.) The chemical processes that occur within simple unicellular ogranisms are, if not well-understood, not a complete mystery. Gap bridged.

Sorry to be dismissive, but to my mind philosophy is what you do when you don't have the intellect for mathematics itself.

15. ## Re: The Mathematical Universe

Sorry but gap not bridged, if chemistry was just atomic particles sticking together it works, but chemistry has the property of self organization, self improvement and self repair these cannot be dealt with by atomic physics so the 'jewel of physics' quantum electrodynamics describes the motion of a single electron in a magnetic field, so there is a long way to go before you can explain the emergence of DNA or it would be the jewel of physics.

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