The limits of mathematics

Thanks, Marcel... and apologies to you and also to Eckard for my mistake in attribution. I enjoyed reading the conversation and learned from both of you.

Marcel, what you say reminds me of Bernard D'espagnat's position when he speaks of a 'veiled reality'. He speaks of an "empirical reality" and an "ultimate reality", and further says that we may say nothing about the ultimate reality. I take it that you think that we ought to be able to say something about it. I've thought, perhaps as you, that what we call our ontology is really a phenomenology, due to the empirical nature of our investigations. I think that what we call "objective reality" is really a consensual, subjective reality.

One problem in saying that something exists in a metaphysical way is the lack of applicability of even the word 'existence'. Heisenberg wrote that even concepts like 'space', 'time' and 'existence" have only limited applicability. How can we say anything at all about this metaphysical reality? Its a tough task, to be sure.

Marcel,

Anton Zeilinger seems to be saying the same as D'Espagnat in http://www.signandsight.com/features/614.html



Q:So there is in fact something that exists independently of us. And the moon is also there when I'm not looking at it.



AZ: Something exists, but it is not directly accessible to us. Only indirectly. And whether this thing must really be called the "moon" is another question. That is also a construct.



Q: But there is something up there...



AZ: ... the word "there" is yet another construct. Space and time are concepts aimed at giving meaning to our world of appearances. So they are entirely reasonable constructs. By no means do I want to give the impression that I believe everything is just our imagination.



Q: The world as a huge theatre that only plays in our heads.

AZ: That is certainly not my view of things.

Q: Then what would you call it, this something that you can't call moon or space or time - this something that exists independently of us?



AZ: Wouldn't I be making another qualification if I tried to give it a name? Isn't it enough if I just say it exists? As soon as you use words like "world" or "universe", you start lugging about all that conceptual ballast again.

Q: But you defend the thesis that there is an "original matter of the universe": information. 



AZ: Yes. For me the concept of "information" is at the basis of everything we call "nature". The moon, the chair, the equation of states, anything and everything, because we can't talk about anything without de facto speaking about the information we have of these things. In this sense the information is the basic building block of our world.



Q: But just now you spoke of a world that exists independently of us.



AZ: That's right. But this world is not directly ascertainable or describable. Because every description must be done in terms of the information, and so you inevitably get into circular reasoning. There's a limit we can't cross. And even a civilisation on Alpha Centauri can't cross it. For me that's something almost mystical.

How will we get past this barrier?

Pankay.

One cannot predict the past but try to understand more or less what happened and what will happen. Animals are often steered by their innate instinct. I understand understanding as a variety of learned memory-based processes in the human brain. While I used to attend some seminars of an Institute for Neurobiology, I regret being unable and not ready to speculate. The more I get familiar with the function of brain, the more I realize understanding almost nothing.

Eckard

Ian,

I did not pay too much attention to FQXi recent, but you did get a very interesting topic of discussion here.

Before answering your questions head on, I want to have a little digression. How does math compare with heuristic arguments? I contend that math proofs and results are like limiting cycles in phase space. A correct proof today is correct forever, while a heuristic argument can either become invalidated over time, or become part of a proof. It is therefore clear that math is timeless and has an independent existence.

But do math and reality have a dual relationship as Gordon McCabe contends? To answer this one has to answer what is reality? Some new mathematics may not have any obvious usefulness in the real world, but this does not mean they are not part of reality. The very paper and ink marks comprising the new math is part of reality. The new question is: do all maths play a role in nature? To answer this one needs to look at examples. What role did crystallographic groups play in the first seconds after Bib Bang? None (or a very minor one). But they become important after first crystals began forming. The moral of the story is that emergence can make any math useful under appropriate conditions. The only question remaining is: is all math playing a UNIVERSAL role in nature? Obviously not. A Minkowski space with 45 spatial and 1 time dimension is not physical.

And now back to Gordon McCabe: reality can describe all maths, and maths can model all reality. I touched above on the first implication. The second one is true if there are not supernatural explanations of reality. Only in that case reality is non-contradictory and can be modeled by non-contradictory mathematical structures.

Florin

" ... maths can model all reality. ... true if there are not supernatural explanations of reality. Only in that case reality is non-contradictory and can be modeled by non-contradictory mathematical structures."

"contradictory reality"? May we suspect reality to be wrong? If a Gordon McCabe contends duality between theory and reality I will never take him seriously.

Laplace admitted to Napoleon not to need the hypothesis god. I feel neither in position nor obliged to have a demon at hand and model anything by means of mathematics. What about hearing I rather trust in physiology. Except for a theorem by Wiener/Chintchine all mathematical approaches so far were rather misleading in this field of science.

Eckard

Rather than talking about an "underlying" order and thus elevating Mathematics to almost an ontological status, couldn't we just say that there is regularity, and it should obviously be possible to derive quantitative statements that reflect the fact there are stable "ratio" relationships. Occam's razor would seem to demand this view, but Platonic ideas are deeply rooted in Western culture, which are not empirically evident nor testable.

In trying to create a quantitative description of the experienced regularity, one has to measure one thing with respect to another, and in this sense nothing stands on its own as an absolute. The small is defined with respect to the large, the large is defined with respect to the small. Inevitably, one gets stuck in circular relationships, circular thinking and thus it will never be possible to arrive at some absolute knowledge, but only relative knowledge. Even counting is dependent on the distinctions we make. We might like to think that these distinctions are "just there", but actually we decide how to break up the whole into parts, and then the counting ensues. This, I think is felt as an unsatisfactory, incomplete knowledge, but it will never be possible to go further than this when "ratio" or the rational method is exclusively utilized.

It is because of the deeply rooted Platonic ideas that terms like "An imperfect understanding" is utilized. Imperfect with respect to what, imperfect in what way? Because it cannot be modelled mathematically in a complete and satisfactory way? The fact that the method known as "ratio" cannot offer an *absolute* description is entirely predictable. But if one is beholden to Platonism, then indeed it would seem "imperfect". But the Platonic view cannot be proven nor disproven... it is outside the method of ratio, and thus "Science"... those who hold to a Platonic view can do so only on faith... there is no place in Science for that, nor should there be.

Either we consider experience primary, or our descriptions as primary. In what sense would a mathematical description be more primary than experience? We can experience order/predictability, but also novelty/unpredictability. Why should this be seen as "Imperfect"? If the world were indeed reducible to mathematical relationships, then we would be describing a machine, not a world of living beings. Why should we expect to understand living beings and the biosphere in an exclusively mathematical way?

Platonism is the idea of an ontologically existing "mathematical realm", and this is related to the idea that the world could "self-assemble" because of these laws. But are these "laws" ontologically existent?

Paul Davies: "the very notion of physical law is a theological one in the first place, a fact that makes many scientists squirm. Isaac Newton first got the idea of absolute, universal, perfect, immutable laws from the Christian doctrine that God created the world and ordered it in a rational way. Christians envisage God as upholding the natural order from beyond the universe, while physicists think of their laws as inhabiting an abstract transcendent realm of perfect mathematical relationships."

http://www.nytimes.com/2007/11/24/opinion/24davies.html?_r=1&pagewanted=2

We should be prepared to say that ultimately, we don't know and won't know. But a more accurate statement would be that by utilizing the method of ratio (including thought itself) it will never be possible to "know". What other methods might we have up our sleeve? it must be realized however that any other method that we might have access to would not be quantitative, but instead qualitative, subjective. This will not replace the method of "ratio" as the type of knowledge gained would be different. It would be unthinkable knowledge that one could have, as mystics, yogis etc. say. To "know" in a intimate way, and not via mental descriptions (mathematical or otherwise) is possible, but this would be "knowing" rather than "knowledge". Of course, this does not replace "Science"... it is instead how we go further and finally scratch that itch to "know".

I agree with the spirit of Einstein's statement: "Subtle is the Lord, but He is not malicious."

Hi Eckard,

"Why do you consider a timespan a measure in an a priori given time between minus infinity and plus infinity when there is no future timespan available? Do you also allow for negative distances in space?"

I don't think that we disagree about time. I think it rather is sometimes difficult to clearly express our ideas about time. I used the word 'in' time. I do not mean to include any future time sense in my use of that word. I agree that the future does not exist. The past no longer exists either. However, we observe events occurring by remembering the past. In that sense we carry a record of the past along with our thoughts. So we say an event took a certain amount of time to occur. We often speak about this time period as if it exists like a rod. We can hold the rod by both ends. We speak about time as if it retains its two ends. This practice is convenient for communication purposes, but it does not capture the essence of time. Time is always now. Nothing takes place in time if time is present only now. However, in our memory time existed for a duration.

We understand meaning from change. Change is always occurring everywhere at every instant. If there is no change, then we receive no information. We draw forth an understanding of meaning from observing change. So I find it convenient, when talking about physics theory, to approach it from the perspective of change. Not just the instantaneous change, but also the prolonged changes that we observed beyond the restriction of 'now' and keep in our memories. In this sense I think there is justification to look among our observations of change, i.e. empirical data, and find something within our memory of time that will give us understanding about the nature of time. I do not see how to do this looking only at an instantaneous 'now'.

In my work I have achieved many good results about the nature of properties of the universe. I achieved some success with establishing that time is absolute. I mean this word only in the sense that neither we nor any other object in the universe can change anything about time. I think the way to demonstrate this is to show that there is a consistently accurate measure of a period of time that never varies. Objects may shrink in size, they may slow down or speed of their processes, however, there is one process in the universe that always occurs at the very same rate. It is a fundamental constant. For me, that universally constant rate is sufficient evidence that time is well controlled, but not by us. I have also found that that special constant rate of time is the key to achieving a unified theory. Its use repairs the damage that Einstain's relative nature of time has done to theoretical physics.

James

Dear Ian,

"I was reading something today that pointed out that the Lagrangian approach to classical mechanics has the position and velocity as the only fundamental variables. I had forgotten that fact. In any case, I wonder if you might be able to use Lagrangian methods to achieve some of what you are trying to achieve."

Thank you for the lead. I will look into it. I have already done most of my work. All of the results are based upon information containing only distance and time. A possible difference between the Lagrangian approach and mine would be that I do not attempt to bring together in a mathematical form the theoretical properties defined by normal theoretical physics. My approach begins and ends with developing results based upon only one fundamental cause. So there is nothing to list or combine together. Everything springs forward from a single common origin. I do identify the cause. However, I can do that only in the same incidental pretend way that today's physics theory does it.

In other words, I do not know what cause is; however, I do know of one theoretical property of the universe that serves the purpose of substituting for the one fundamental unknown cause. So it is known theoretically. I am interested in the possible treatment of using only distance and time in the Lagrangian method. I will look back and see what was done. I probably will not be able to use it in its own form; however, it may help me to see how to clarify my own ideas for others. In any case, it may finally represent something that I can use as a reference to point to. Truly original work does not seem to give much opportunity to point to others for help in making one's case.

James

Dear James,

I refer to influences in the moment of consideration. Independently of any subjective memory they definitely relate only to the past.

Regards,

Eckard

Ian,

You are right. And the future possibility is unrevealed today because mathematics in inexhaustible. Thinking outside the box about emergence, I believe no where but in humor we see this in its extreme form. Humor is very contextual and extremely sensitive to the natural selection rules of the sociological "landscape" (to paraphrase Susskind). For example, to people 100 years ago, a Seinfeld episode would looked completely strange. We laugh at different things because we constantly find new patterns in our daily life. And those patterns are just new emergent mathematical axioms. But this is maybe all too fuzzy. We can pick another area: morality and law. We can understand morality as an ever evolving landscape of fuzzy heuristic possibilities and the new laws as quasi-formalized axiomatic systems that passed at least some form of consistency check.

Florin

Ian,

When I mention the idea of a single fundamental cause, I am not referring to boundary conditions or an initial act such as the big bang. I am thinking in terms of a single, ongoing, always present cause for all activity at all times. That is why I pointed, in an earlier meassage, toward the importance of properly interpreting even as simple an equation as f=ma. If it was properly interpreted, then the cause for all activity would reveal itself even in that equation.

It cannot happen under current circumstances, because, the equation was very quickly made subservient to unempirical ideas such as mass being an indefinable property. Once that decision took hold, the equation became incapable of revealing any information that could help us identify a single cause for the accelerations of all matter under all conditions.

Instead what happens is that theoretical physics introduces additional indefinable properties that lead to different mathematical identities for f. In other words, force cannot reveal information about a single cause because force is theoretically separated into gravity, electric force, weak force, strong force, etc.

Each indefinable idea such as mass, electric charge, temperature, other than distance and time subjects the equations of empirical physics to artifical restrictions without empirical justification. In other words, the added on ideas of theoretical physics distort and restrict the usefulness of the equations. The empirical form of the equations, before the theoretical ideas weigh them down, are the most useful form for the equations to exist in. It is in their empirical form that they can be inspected and contemplated and reveal real knowledge, with real empirical support, about the underlying cause for activity in the universe.

I recognize the incredible mathematical skills, vast theoretical knowledge, and empirical knowledge of theoretical physicsts. I do not wish to appear to disrespect their high level of education and intelligence. However, I am pointing to fundamental theoretical decisions that remain, in my opinion, unsubstantiated and unexplained to this day. I would argue that higher level theory, built upon a foundation of unexplained theoretical ideas, can be no more correct than are those fundamental ideas. The errors are likely compounded as theory becomes more and more complex.

Here is a different example of an unexplained fundamental property. I have raised this issue before in earlier messages in other forums here: What is thermodynamic entropy? I am not asking for an explanation that skips past a direct immediate answer and moves instead immediately to Boltzmann's interpretation and others that followed from it to the present day understanding of available microstates. Thermodynamic entropy is successfully used for important work in different fields. It is something real and yet we cannot explain what it was that Clausius discovered. He gave a very precise definition with its ideal conditions. Those conditions must be met or the calculation of thermodynamic entropy will be incorrect.

These conditions do not apply to the alternative definitions of entropy. In other words, thermodynamic entropy is not represented by those other definitions. What is thermodynamic entropy? Furthermore, its definition includes temperature. Temperature is another indefinable property. What is temperature? I am not asking what it can be related to in so far as related changes occuring to other properties such as energy. Temperature is not energy. I do not think that kind of answer explains what temperature is. How can these questions go unanswered to this day? Why were they skipped over? What problems has this caused for higher level theoretical analyses? Are they fundamental evidence that the mathematical equations of 'theoretical' physics, not empirical physics, became separated from reality right from the start?

James

Ian,

I decided I should make it clear that I am not myself searching for these answers. I have answers. I am asking for answers from theoretical physicists. I am avoiding filling this forum with my ideas, yet, I think it is important to raise these questions and establish whether or not they have been answered or can be answered by theoretical physicists. I say they definitely have not been answered directly or explained as being clear separate properties in their own right.

James

James,

You wrote: "Theoretical physics has been speculative right from its start" and "I am not referring to boundary conditions or an initial act such as the big bang. I am thinking in terms of a single, ongoing, always present cause for all activity at all times".

Given, there is no understanding of a ubiquitous cause available, why should science reject sometimes using speculative heuristic elements in addition to predominantly success-controlled methods? Let me return to what Ian solicited, discrepancies between speculative mathematical physics and what is reasonably considered the real world. Can you contribute anything but sterile skepticism?

I am not sure whether I already uttered my opinion both understandably and acceptably enough: We should search for overlooked cases of non sequitur. Why did you simply ignore my arguments instead of trying to refute them? I wonder why we seem to agree so much with respect to speculations like Big Bang and boundary conditions.

Eckard

Eckard

I have a strong evolutionary standpoint on mathematics. I consider mathematics to be nothing more than a communication product of the evolution of human brains, which means that understanding mathematics first boils down to understanding human brains.

I speculate that brains started with the macroscopic multicellular organism's need to move around its environment to reach better living conditions. To this mean, it developed a motor nervous system to innervate muscles for contraction. It then developed a variety of sensory systems, feedback mechanisms that, once appropriately engineered and linked to the motor component, enabled movement adaptability to the changing surrounding environments.

I'd argue that ever since the sensorimotor system has been developed, the brain's evolutionary challenge has been to determine what is the sensorimotor algorithm that provides the best chances of survival (say, for a given species in a given environment). We started with nervous arc reflexes (motor and sensory neurons); we then developed more sophisticated neural computations that required more and more interneurons to process the computing steps between the motor and sensory neurons.

As our brains evolved in an environment to which they were sensitive, they were exposed to recurring sensory patterns of "physical phenomena" and their evolution was shaped by conserving the neural organizations that most accurately predicted these "physical phenomena" for the largest number of contextual sensory inputs. These neural organizations were the rudiments of logical reasoning.

This means that human logical reasoning has been shaped based on the "physical laws" that seem to permeate our external world. This explains why a Cro-Magnon individual could mentally calculate the angle and force that would throw his spear to hit its target: its brain had internalized the physical laws of ballistics to which it had been repeatedly exposed to. This was way before mathematics was developed. Then, one day, some individuals started transcribing on paper these principles of logical reasoning in their best possible accuracy, and this was the birth of mathematics.

Mathematics is thus not an invented science nor a discovered science. It is a means of communication of logical arguments between individuals. It is the human logical reasoning system externalized into a universal language.

From then on, things started speeding up. Logical reasoning segments could be communicated with great speed and efficacy between individuals, and soon a whole community could work on the same logical reasoning problems that in the past, individuals had been trying to solve alone. This increase in total computing resources enabled human beings to solve more and more complex logical reasoning problems, which, because the logical system was tailored to reflect the macroscopical physical world that had surrounded our brains throughout their evolution, were proved to be very useful in the macroscopic world. Protocols of conduct were elaborated on the basis of this successful reasoning system, one of which was the scientific method, which has led us to the kind of science we know today.

As said earlier, mathematics has proven to be very useful at understanding macroscopical physical problems (e.g. problems in Newtonian mechanics), which also seem intuitive to "understand" to the human brain. However, we must keep in mind that mathematics is actually a reflection of our logical system, which was designed to work in a macroscopical physical environment. When we try to use mathematics to describe the physical world at different levels, i.e. at the microscopic scale, the nanoscale, or the cosmologic scale, things get tougher and we find ourselves with logical conclusions that seem more and more unintuitive and even far-fetched. I think this is because our logical system is not adequate to describe the external world on such scales.

Therefore, I think that, like any other evolutionary tool, mathematics may very well have use limitations, especially when used to describe physics in the non-macroscopic scales. Things in mathematics that are not physically conceivable, such as the concepts of a point or of infinity, are not a problem for me, as I see these as simply some of the basic concepts that our logical system uses to base its operations on.

The question is, will this logical reasoning system be sufficient to enable us to describe, within this system, the physical phenomena on all possible scales? It may or may not be, but I think that the most likely scenario will be that living beings will eventually use a superimposition of many evolutionary tools (some of which may not yet be developed) at once to solve complex problems of description such as the ones we currently encounter in quantum physics and cosmology.

I haven't specified why I put some words in quotes, nor have I touched on other topics, esp. truth (I think there is no absolute, Platonic truth, and that our tendency to relentlessly search for one truth is part of our pre-programmed logical reasoning system), but if you are interested I would be glad to share these thoughts too. I would be very curious to hear about your opinions on my evolutionary pure speculation!

Eckard,

I apologize. These last two weeks were uncomfortable for me to sit at my computer. I am right now printing off messages and I will reread them. I may have missed reading or did not read carefully enough some of the messages. I will look back over the messages of these last two weeks and prepare a response.

James

Florin,

You wrote: "the future possibility is unrevealed today because mathematics in inexhaustible."

You are definitely the son of exactly one mother and one father. Maybe, you already have children and possibly even grandchildren. This is not revealed to me. However, I doubt that their grandchildren already exist in the sense they are merely not yet revealed because mathematics is inexhaustible.

Can we really just not see in advance future events like a clairvoyant or did they simply not yet happen? I imagine the moment of consideration like a border of integration for all influences from reality. Some influences may be calculable others are more or less likely, some are seemingly marginal, unseen, or erratic; but the origins for all of them belongs to the past while seen from moment under consideration the future integration is as ramified as is the causal tree of the past. I do not deny that there are reasonable heuristic rules, e.g. trees do not grow too high or mors certa.

Is something wrong in this consideration?

Eckard

Eckard,

The point I was trying to make was not that future exists because math is infinite, but that an interesting and novel future exists because of that. Over time, many people proclaimed "the end of history", or equivalently that everything is already discovered. The inexhaustibility of math is what keeps things interesting for future generations, otherwise we would be conceptually no different that a hamster going round and round in its wheel.

Florin