The limits of mathematics

James,

You wrote:"The t in physics equations is a measure in time (some duration of some physical activity) and not of time (as in a testable property in itself)."

Zeh also dealt with the question of "direction in time" (A) or "direction of time" (B). He called his book "The physical Basis of the Direction *of* time.

I feel forced to almost agree with your statement because you refer to the common assumption of an a priori given coordinate time (A) for using "t in equations" (C).

Nonetheless I would like to object that this use (C) refers to an abstract notion t that cannot be measured at all. Let me call it even unrealistic in so far as future is predictable to some extent but it definitely evades measurement in advance.

Any clock primarily performs a measurement *of* currently elapsed time, not *in* elapsed time. You have to synchronize it before it shows a moment in commonly agreed time. Notice, I changed the point of view. The usual notion of time needs an arbitrarily agreed point of reference. Positive elapsed real time counts "backward" from t=0. The same holds for anticipated elapsed time.

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?

Eckard

I blamed Dirichlet and Dedekind wrong. Let me start explaining. There are several problems concerning definitions that were made for R if we try to follow physical restriction to R+, cf. e.g. criticism by Terhardt of integration for unilateral Laplace transform from a small negative value to +oo or by Aseltine §3-6. Likewise, the usual illustration of Dirac impulses fails with restriction to R+. Common sense tells us that each single rational number must be countable in the sense it is different from any other one by a non vanishing value while genuine continuity as defined by Peirce demands that real numbers behave differently.

Dirichlet ignored this different properties when he gave the well known example of a function f(x) that is 0 for rational x and 1 for irrational x. While it has been called nowhere dense, I would rather see this the other way round.

Furthermore, Dirichlet attributed the mean value [f(x-0)+f(x+0)]/2 to discontinuous functions. Using pertaining integral tables I found out that the mean value is misleading. It does not make sense to exclude or include a "single" real number from the continuum of reals. It does not matter whether or not zero, infinity or any rational number exist within the continuum. There are infinitely many possible substitutes that cannot be distinguished from it. Accordingly I consider |sign(0)|=1 justified and a separate treatment for the very point zero between R+ and R- pointless.

Dedekind continued the strictly speaking inadmissible while pragmatic neglect of the fundamental difference between rational numbers and the uncountable manifold of potential points of continuum. He caused additional confusion when he referred to numbers as to points instead of measured that relate to a common unity.

Cantor caused a tempest in a teapot and ongoing trouble when added some useless speculations and the analogy based notion of cardinality. Isn't it sufficient to distinguish between in principle countable, discrete, rational numbers on one side and fictitious real numbers of continuum on the other side? I never heard of any application for aleph_2. Of course, I distinguish between let's say the number five as a rational number and an equivalent real number five.

Eckard

Ian, there can never be totally isolated physical objects outside of mathematics?..for instance an absolute nothing, will have a discrete part of it as a finite of "something"?..and thus likewise a total everything something(infinity)..will have a "zero" assigned part to it?

The Riemman Zeta Function for zero distribution is really natures mathematical devise that orders zero's to this effect?..at least this is my understanding!

In this sense mathematics itself has to adhere to the same fact?

best p.v

Eckard,

I think you raise good challenges, and I am glad you are actively participating here. I will not be answering quickly, because, I have had a busy week that led to my undergoing an out-patient surgery today. I am back at home, doing fine and it is not real serious. However, I think I will wait until I am off pain medication before responding. I just wanted you to know that the delay is not due to lack of interest.

James

Dear Eckard,

How do we get back to knowing, rather than doing.

Pankaj,

You wrote: "How do we get back to knowing, rather than doing."

-- As someone whose mother tongue is not English, I have to admit being not sure I understood what you meant in your rather cryptic question without question mark.

Some experts argue not just the future but also the past is unknown to us. I do not think so. Well, it is imaginable that somebody lost his memory while he can be told what will most likely happen. My reasoning refers to physical reality which is exclusively determined by influences that originate in the past. So it is reality that does not know the future.

Does this answer your question?

Eckard

Eckard,

I didn't even realize that I had placed a . instead of ? until I read your comment... nothing cryptic meant by that. I had read through this thread and found your point of view quite compelling, I found myself very often agreeing with you. I was curious about what you meant by "It is a choice we made long ago between knowing and doing.". I thought that I could learn something more from you, so I asked what you meant by that.

Something more specific then... when and how did we make the decision towards doing, rather than knowing... and who made this decision ( I mean of course in historical terms, and not necessarily focused on one individual)? I agree with you that prediction is not the same as understanding. What then would constitute understanding? How could we move towards understanding? Perhaps you will write something on these matters and I will understand some more of this world.

Thanks,

Pankaj

Pankaj,

I am actually the one who wrote this:"It is a choice we made long ago between knowing and doing."

Knowing comes having a logical explanation to support a mathematical description. A logical description is about really understanding the physical laws we use to describe the various behaviors of the universe.

How do we get back to "understanding"? We may do that by admitting that something does really exist by itself in a metaphysical way and by deducing the logical requirements of such existence. Nothing in empirical science admits that something exists by itself. The empirical approach only deal with our experience ....

Hope this helps,

Marcel, ( I am still around!)

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