The limits of mathematics

Ian,

Anselm of Canterbury subordinated knowledge below belief. While I got the impression that FQXi includes a lot of believers, I would rather like to follow Otto de Guericke who preferred experiments that led to the steam engine and to investigation of electricity. Moreover I grew up in an environment of horrible consequences of false ideologies where I early learned to distrust. You moved from mechanical engineering to theoretical physics. I admire this. Maybe, I more often felt confronted with details that seems to be not quite logical in applied mathematics. Meanwhile there seems to be mounting evidence that my suspicions are not unfounded. Let me add my guess: Even if the world is endless as Nicolaus Cusanus (1401-1464) explained, which contradicts to the Big Bang and Genesis, it is the only reasonable point of view for physics to restrict their scope. Restriction to analysis of reality means restriction to what is past relative to the moment under consideration. Such restriction must not be omitted in the mathematical models.

Eckard

Dear Ian Durham,

Here is an excerpt from an essay I wrote. The basic idea is that our only source of empirical knowledge is via photons that carry information about changes of distance with respect to time. My premise is that all properties should be expressible in terms of these two empirical properties. In other words, all theory should be clearly traceable, through their adopted units of measurement, to empirical evidence consisting only of measurements of distance and time:

...Our mathematical equations contain initial conditions represented by numbers and units of measurement, and final conditions also represented by numbers and units of measurement. So, it is the numbers and the units of measurement that must be made right. The numbers are not the source of our theoretical problems. The meaning of the equations, insofar as mathematical theoretical value is concerned, is carried through the establishment of units of measurement. It is important that the meanings represented by units of measurement are meanings that reflect the real nature of the universe.

The units of f = ma are crucially important for the rest of physics theory, and they must not be based upon ideas that are guesses. The units of force and mass must be defined by means of a direct connection with the two units of measurement by which all empirical knowledge is made known.

Our empirical knowledge is filtered to us through measurements of distance and time. For this reason the units of distance and time are fundamental and do not have to be defined. They simply are chosen. In the equation f = ma we can rely upon acceleration to reflect the natural operation of the universe. Insofar as the units of mass and force are concerned, they must be definable in terms of distance and time or we are lost at the fundamental level. How can units of force and mass be formed from those of distance and time? We need first to discover how to define one of them. The second can be solved in terms of the other.

First we determine a connection that seems reasonable based upon non-theoretical knowledge. If we solve for f/m = a, we see that the combined units of force and mass must be reducible to those of acceleration. In other words, the replacement for newtons divided by the replacement for kilograms must reduce to meters divided by seconds squared. In addition, we can expect that they must have a reasonable chance of making physical sense. All physical sense is made known by changes of velocity.

We can try the guess that they have units of velocity or even guess they have units of change of velocity. There is also the possibility that one of them does not have units. As a first effort at solving this problem, we may begin with the simplest possible interpretation for force. This interpretation is that force represents a ratio of two quantities of the same property. This would define force without units. It also assigns units of inverse acceleration to mass. If this solution is real, then force may be the ratio of object acceleration to a more fundamental acceleration.

If this kind of relationship could be established, then what we would be looking for is a pervasive form of acceleration that becomes traded off for object acceleration. It would be a matter of conservation of acceleration. Unity requires that we avoid concluding it is a mysterious new property of the universe. We can expect it to be an acceleration of a single, most fundamental, property.

To where would such a radical change lead? There is a great deal of successful theory that would also have to change radically. Almost everything would have to change. The operation of the universe would remain the same, but its theoretical interpretation would be redeveloped. However, if the choice is correct, the result should be very rewarding.

Fundamental truth will certainly outperform artificial theory. Two, far from ordinary, results of following the path suggested here are:

h = kec

This says: Planck's constant is numerically equal to Boltzmann's constant times fundamental electric charge times the speed of light.

f = kw

This says: Photon force is numerically equal to Boltzmann's constant times frequency. The change of units for force and mass, made as described above, leads to these equations. When these changes are made, the units for these equations match. They are evidence that new knowledge is possible at the fundamental level. They are evidence that a great deal of theoretical revision is in order. ...

James

Thank you for your reply. Let me take this one point at a time. I need to understand why it is that you imply that sound and touch are not the result of the action of photons? If I misunderstand this point, then I fundamentally misunderstand.

"...I disagree with you on this point since that seems to assume that our empirical knowledge of the universe is confined to those sensory perceptions that involve photons, but (ignoring how the signals get from our sensory organs to our brains), we can learn quite a bit from sound as well as touch (maybe even touch). ..."

James

James

"...Unless, of course, by photonic you mean that all sensations are somehow traceable to electromagnetic interactions. I would still disagree with you on this, though, since there certainly are some that are traceable to gravitational interactions..."

That is what I mean. Gravitational effects are included; but, I am not yet explaining gravity. I think I have done the necessary work to explain gravity; but, I would like to concentrate on the idea that particles of matter interact by photonic means. I am saying there is one single fundamental cause for all activity. I am also saying that the evidence for this singular cause is demonstrated in the effects. The effects are always detected as changes of velocity.

I discount theoretical ideas such as the existence of gravitons, or even Higg's particles. That is not necessary to prove at this point since gravitons and Higg's particles are not yet discovered. I am speaking about what is known now at the most fundamental level. I think that it is correct to say that all effects known are the result of the interaction of photons between particles of matter. The idea I put forth as to a possible nature of mass is the main point that I wish to make. I can go further with this idea.

Thank you very much for responding. I will move onto your second point if it is agreed that sound and touch are, strictly speaking, the result of photonic interactions between our particles and those of the instigator particles. I am not yet saying what I think the cause of gravity is. I have only suggested that mass be connected to empirical evidence as we know it.

If mass is introduced as a new undefined theoretical property, then, I think, that is something to be discussed further. Justification must be given. Lack of understanding is not sufficient reason for introducing a 'given'. That is what I think. Everything introduced should be securely anchored in empirically observed properties. I think those properties consist only of changes of distance with respect to time.

Please continue to respond with your critiscisms. I appreciate it. I am truly interested in uncovering truth even if it is not mine.

James

I need to amend something I said. I should have made it clear that, so far, I am speaking only about atomic interactions and not nuclear.

James

Ian,

you solicited thoughts on " ... the Quine-Putnam indispensability argument which basically says that if we believe in the concreteness of the physical theories described by mathematical objects then we also ought to believe in the concreteness of those mathematical objects themselves."

"Concrete" as opposed to what? Abstract? Then is there such a thing as "concrete" language? That is, do symbols stand for themselves only, or are they independent of the objects for which they stand?

My view: language is independent of meaning. It follows therefore that physical (mathematical) theories are no more concrete than the language (mathematics) that describes them. There is a distinction to be made between objects and meaning; there is no such distinction between a (physical) theory that maps symbols to symbols and a (mathematical) theory that maps symbols to objects. In other words, physical objects and mathematical objects are both symbolic representations independent of phenomenological observation. Thus, physical science is necessarily an open system, progressing toward what Popper called verisimilitude, an asymptotic approach to truth, and never offering a completely closed judgment of truth, a proof of its conclusions. Mathematical science necessarily offers closed judgments based on axiomatic deduction, and theorems (true mathematical statements) are proven in the domains to which they apply. The physical domain--being the whole observed, and even perhaps the unobserved, universe--may not be subject to such axiomatization. How would we know in any case?--Goedel taught us that no set of axioms is sufficient to prove its own self-consistency; there always exist true statements that cannot be deduced from the axioms.

That being said, there are serious attempts to recast mathematics as an experimental science. Chaitin, Wolfram, et al, may really lead us to a common point of closure between what we say about the world and what the world says back.

Tom

Dear Ian,

This is what I think: We only know about effects because that is what empirical evidence consists of. We do not know what cause is. It is our lack of understanding about the nature of cause that makes theories necessary. The theorist imagines what causes they think could be responsible for the effects they see. I say this because photons causes changes of velocity. We do not know why they do this, so, we imagine that they have properties that we interpret to be causes.

When I speak about our learning about the operation of the universe by observing patterns in changes of velocity, I mean that that is all we ever directly observe. I avoid inventing indefinable properties. I also avoid inventing causes. Names of convenience are fine to use. What is not helpful is to force our theoretical guesses about causes into empirically determined equations through the means of indefinable invented units.

There is no escaping the necessity for one unknowable fundamental cause. Saying that it is unknowable is not meant to suggest that we cannot identify it and name it. Unknowable is only intended to acknowledge that it is a given without explanation of its origin or ability to cause action. One miracle such as this is enough. Anymore givens are just taking us further away from recognizing that fundamental unity exists right from the start of the universe and this should be reflected in theoretical physics. Fundamental unity should be a part of physics theory right from its start.

I am attempting to make just two points at this time. One is that all empirical information is gathered in the form of patterns of changes of velocity of objects. The second is that I free mass from its artificial theoretical definition, I do not do away with it. It remains as the m in f=ma. All that has changed is that that simple equation has been returned to its empirical roots. The changes that follow because of this act affect almost all of physics theory.

I will not use my own theoretical work to demonstrate this by bringing those ideas here. This forum is not the proper place for me to expound on a new theory. So, I will instead see if I can write a few messages that stick to this idea of returning all physics equations to their empirical roots. I can touch on ideas such as energy and momentum.

James

Ian,

You wrote, "So how much of what underlies mathematics is 'real?' Since much of mathematics can be described as a process, are these underlying processes real?"

In those terms, I think we _have_ to take mathematics as an experimental science. That is, as you implied elsewhere in your post, there is no reality in arbitrarily chosen symbols and their manipulation; as an art, mathematics is not "about" anything, any more than natural language--that the syntax of a statement may be entirely correct, does not imply meaning.

If we take meaning as real (that is what I mean by 'what the world says back to us'), then the meaning that comes from a computer programming language is as real as the substrate on which the program runs, because that is the process, the mechanical throughput, that mimics all physical processes. I.e., information flows continuously over the substrate, though the exchange of information between nodes is in discrete units. The bad news is that Chaitin has discovered an algorithmically defined but uncomputable number (Omega) whose digital expansion is normal but whose value depends on the computer language running the algorithm. So even our arithmetic, in this context, possesses a degree of built in uncertainty. This raises the question of whether a complete one-to-one relation between language and meaning is even possible.

Tom

Dear Ian,

That is a undergraduate textbook. I do not have it; however, I have formally studied physics and have many books. I was just trying to introduce a different way of deriving theoretical physics. It does not do away with any conservation laws. Theory must adhere to empirical evidence. What it does do is remove all of the, in my opinion, artificial theoretical guesses that are preventing us from achieving unity without having to add on more invented artificial theoretical guesses. Unity should not be an after thought that must be forced onto theory by introducing additional unverifiable properties. Anyway, I have thought a great deal about this and have completed a great deal of work demonstrating it. I think you have been extremely polite to me. I appreciate your interest. Maybe this isn't something to pursue here. It definitely does not embrace the approach to physics the that book teaches.

James

Dear Ian,

No I am not affended. You are as nice a person as I have met here. It just that the question had to do with discovering when math no longer is representing reality. I have studied and used theoretical physics. I am not a physicist, but I think there is something important to be pointed out about what physics theory really represents. The difficulty to overcome is that most theory is believed to be reality. I was suggesting that none of theory represents reality. When I mentioned that the definition of mass could be chosen differently, I think it should be clear that the original choice made was a guess. No one could possibly know that mass deserves to be an indefinable property. When I mention that we learn everything via photons and that their information is limited to observing patterns in changes of velocity, I don't think that that is a theoretical statement. Photons begin with changes of velocity and end by causing changes of velocity. We can theorize about what photons hold or contain that allows them to cause changes of velocity; but, we cannot know that. It is theoretical.

My point from beginning to end is difficult to make clear against the common belief in theory. This is my point: The best 'theory' is the one that removes all theory. Theory helps us to keep our thoughts straight, but it does the opposite in terms of learning about reality. I do not mean that it is not useful. I only mean that it is invented as a means to proceed with analyzing physics knowledge without needing to understand the nature of cause. We do not know what cause is. Everything that is attributed as being a cause is invented. This practice of theorizing may be useful, but in terms of understanding the nature of the universe, it is misleading and at times misguided. If we wish to allow math to truly serve us in learning about the nature of the universe, then we must let it take its own course and not steer it this way and that way by interjecting theoretical ideas.

The difficulty with suggesting this kind of approach is that theory clearly stands in the way. It is believed to be true. How can one say that everything could be changed when it is believed that everything is known to be true? Anyway I work for the change. That is why I work alone. That is why I put my work on the Internet. There is no other way to let it out. There are no other books or sources to point to. I am not insisting that others should quickly recognize that I am correct. I may not be correct. However, I think it should be possible to challenge theory without having some theory put forward as evidence of the correctness of other theory. I think that theory can never prove theory. The mathematics will work out properly, but the mathematics has long since become the tool of the theorist and the servant of theory.

James

James

Ian,

There are plenty of technical primary and secondary sources available; however, I think the most comprehensive view and most fun to read, though philosophical and not technical, is Chaitin's collection of essays:

Thinking About Godel and Turing: essays on complexity, 1970-2007.

Tom