The limits of mathematics

Hi Ian,

I'm not exactly sure what "divergence" you're asking about here... Starting from the reasonable premise that pure mathematics is not part of the physical world, then the question isn't when they diverge, so much as when they can be said to be vaguely analogous. The branch of physics that deals with the rough overlap between math and reality is measurement theory (which tells one how to map physical events onto mathematical structures and vice-versa). And of course there are an infinite number of (presumably incorrect) measurement theories where the math diverges from the physical reality.

But still, I think I have a rough idea of what you're asking... so consider these two examples which might help clarify the issues you raise:

1) When you solve the (time-independent) Schrodinger equation for the Hydrogen atom, you get a continuum of solutions that just happen to mostly be unnormalizable. It's the measurement theory -- which demands normalization to make physical sense of the solution -- that effectively "eliminates" most of these solutions on grounds that they're not physical. So without this "divergence" between math and reality, it's not even obvious that quantization would emerge in the first place. (One can run across many examples of how physical reasoning impacts the "allowable" mathematical solution space in just about any field of physics.)

2) In quantum measurement theory, one does not map the mathematical wavefunction directly onto physical reality, but rather the "probability" of various real outcomes. Still, once the measurement is made, the probability of the actual outcome jumps to 100%, meaning that it is perfectly acceptable for a (mathematical) superposition to get mapped to one particular physical outcome. But quantum measurement theory also governs state preparation -- via a non-probabilistic rule. Here the particular physical outcome is always mapped onto a mathematical pure state, never a superposition between that state and others. So the map between physics and math is now asymmetric; one uses one set of rules when going from reality->math, and another set of rules when one goes from math->reality. If this asymmetry is real, then the premise of a 11 equivalence map between math and physics is demonstrably false. (For the record, I'd rather keep the math/physics map, and symmetrize quantum measurement theory...)

Ken

Dear Ian Durham,

I presume that the question you raised is intended for Professional level discussion. I am not a physicist. My first message was a general statement of my opinion that the use of mathematics by theoretical physicists has been only loosely connected to reality. It is true that the equations make excellent predictions. This is as it should be since they are modeled to fit the patterns observed in empirical evidence. It is the patterns that do the work of leading us to successful predictions.

Where I see theory and reality separate is in any attempt to invent indefinable properties and their units. It is through those invented units that the invented properties become solidified into the equations. Before this act is commited, the equations accurately represent empirical knowledge. After that act of inventing properties is begun, the equations veer away from empirical knowledge and become subservient, even captive, to theoretical ideas. If any added properties and their units of measurement cannot be defined in terms of the original empirical evidence, they are guesses.

I will end this message here. There were no comments made with regard to my first message. So, if this line of thought is not relevent to what you want to explore, then I will not pursue it. I have re-read your essay in order to better understand your viewpoint. Congratulations again on winning a prize and membership.

James

Dear Member Ian Durham,

When did mathematics "diverge"? You might take issue concerning any opinion I uttered for instance in my essays 369 and 527. Being a retired teacher of fundamentals including complex theory of signal processing to students of engineering, I am trying to shed light into several related matters.

My basic argument was: Future data are not available. There is no reason for always performing a spectral analysis by means of complex Fourier transform with integration over time from minus infinity to plus infinity. Instead, cosine transform works well in practice too. However, some theorists considered and perhaps still consider Fourier transform mandatory "for mathematical reasons".

Also "for mathematical reasons" students have to learn |sign(0)|=0. As shown in my IEEE paper, use of such singular value may be misleading. I mentioned just this example out of many flaws because I recently found out who is to blame for an overlooked deviation from traditional meaning of number. The idea of 1858 was not published before 1872. At that time all mathematicians including Weierstrass still followed the definitions in Euclid's Elements, Book VII, 2: "A number is a multitude composed of units". In terms of the number line the unit is a measure of length, the distance between two points, a reference point zero and an endpoint. Richard Dedekind intended getting rid of geometry and restricting the fundamentals of calculus to a purely arithmetic basis when he attributed the numbers no longer to the distances but to their endpoints. This means a tacit replacement of one-dimensional measures by zero-dimensional points of these pieces of the line. We may interpret any replaced distance between zero and an endpoint as the integral over a Dirac impulse at the endpoint. Even if Dedekind insisted of the distinguishing between his cuts and the numbers they created, he ignored the role of the unit "one", and he also ignored that a continuous line can endlessly be split in parts while a point has no parts at all. Meanwhile, the theory of point sets has incorporated Dedekind's identification of numbers as points and has become a gospel-like tenet in mathematics despite of never really resolved paradoxes.

The introduction of a neutral element each for addition and multiplication restored the unit to some extent.

Why is topology still unable to perform a symmetrical cut? For instance, the mathematicians have to arbitrarily decide how to cope with the very point zero between positive and negative real numbers.

Let me say first that I consider mathematics something to discover like the interplay between pi, e, and i. As long as there is room for arbitrary definitions, axioms, etc. the natural relationship has not yet been found.

I prefer the good old and perhaps natural attribution of numbers to measures, not to points. A continuous piece of a line extends between two points without including or excluding these points. I have no idea how to single out any real number from a continuous line on a not insane basis.

Bijection between measures along a first line and a second one should not link point to point but the limits from inside the piece to each other. Singular points, lines, and areas are accordingly unphysical in general. Point charge, line current, etc. are valuable fictions.

Do not get me wrong. I just focused on comparatively benign deviations already inside mathematics in order to not loose all support from physicists. As soon as we agree on them I could risk dealing with more virulent matter.

Eckard

Anonymous was me, Eckard Blumschein who offers alternative explanations for a few oddities that are bothering the mainstream. Since you Ian Durham meant "infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno." I would like to add that Zeno of Elea (490-430) was a disciple of the monist Parmenides of Elea (515-445) who denied time, motion, and the split of the continuum into points. Zeno's paradoxes relied on missing ability to distinguish between abstracted ideal notions like point, line, number, infinitely divisible continuum and the obvious reality. You are quite right: Continuity implies infinite accuracy. In other words, both ideal ideas (continuity and infinity) are likewise unphysical. For instance, E(R=0) of a point charge is mathematically infinite. In so far, mathematical models diverge from reality from the very beginning. However I do not consider this the crucial divergence.

While I imagine you will need some time as to read a few times and scrutinize my essays and the added files, I would eventually highly appreciate your reply.

Eckard

Physical evidence of numbers? I see such kind of articulation close to blown up points, rotated points, 3D points, and the like. Isn't the design of my home quite different from the really existing home? I do not refer to disagreements but to the fact that a plan is always limited to the realm of imagination while my home exists in reality.

I would like to distinguish between unavoidable abstraction and unjustified additional gaps between mathematics an reality. These gaps arose due to the fallacy that widening of mathematical notions implies enlarged degrees of freedom in physics too. In particular I blame the unrestricted unphysical use of negative and complex numbers. People in pubs describe mathematics as something ridiculous where three out of two people left a room and therefore someone has to come in as to make the room empty. Ask your laser colleague which of the two alternative complex representations he considers the physically true one. Both together is impossible, and so far the decision for one out of the two is arbitrary. Already the engineer Raffael Bombelli (1526-1573) understood: Any complex number always occurs together with its conjugate (col suo Residuo, cf. Gericke's Geschichte des Zahlbegriffs, p. 61). When Gauss in 1831 promoted complex numbers by representing them orthogonally in complex plane, he omitted the conjugacy.

No expert questions the usefulness of complex calculus. We should deal with the question whether or not final complex representations are really indispensable. Engineers do not have scruples when they e.g. use the complex relationship between two quantities. However, they learned to transform single quantities back into reality where the imaginary numbers vanish. Physicists tend to neglect such trivialities. The team Erwin Schroedinger and Peterle Weyl did not obey the due obligation of correct transformation into complex domain and return.

It might be a bit exaggerated when I am nonetheless calling Hermann Weyl the last at least consequently and honestly thinking physicist when he uttered "at the moment is no explanation in sight" and later "we are less certain than ever about the ultimate foundations of mathematics".

Eckard Blumschein

"Many thanks to everyone for their thoughts (and keep them coming!). Ken: (good to see you checking these forums, by the way) since you began from the premise that mathematics is not part of the physical world, I counter with the Quine-Putnam indispensability argument: 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."

So, if I believe in quarks, I have to believe in sets, is that it?

What about applicable (useful) math that is inconsistent? Does this mean that you believe in inconsistent math?

It is possible that you have to have a sort of Platonist mindset to come up with some of the outlandish math that is around, however that doesn't make it right in an applied context and certainly doesn't make it real.

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