The limits of mathematics

Dear Member Ian Durham,

"...What I am interested in from a non-speculative standpoint are those places that mathematics *diverges* from the physical world, i.e. those places in which mathematics describes something that is very clearly not physically possible. ..."

I understand that your question probably addresses advanced areas of theoretical physics where even theoretical physicsts might openly question the physical meanings of their mathematical expressions. So much of what I see discussed in the Blog section already strikes me as having long since left reality behind. It makes me wonder: How can we know when theoretical physics has gone too far astray? I think it happened a great many moons ago. Many theoretical physicsits appear to me to believe that they are still well within the bounds of reality. It is as if internally consistant mathematics is the test for reality.

My opinion does not fit within the boundaries of your inquiry. I think that as soon as theoretical physics began to guess about the nature of cause, they went astray. I don't know yet how to prove this. The only avenue I see available is to show that fundamental theoretical unity is possible right from the start of development of theoretical physics. Afterall, if the fundamental causes of theoretical physics are not necessary to interpret empirical evidence, then perhaps theoretical physics has been chasing its own tail. This approach hasn't been received well. Yet, when I see the many exotic theoretical paths that theoretical physics has veered off into, I wonder: How did things get so far? Anyway, you now have at least one response. My response could be dismissed as being speculative. My response to that is: Theoretical physics has been speculative right from its start.

James

You have a mathematical model that has been derived under some formal system.

Any model of anything is just that, a model; it will include some assumptions, some explicit, others implicit.

It may or may not represent reality or more likely it is an approximation to some degree.

The creator and/or user of the model must decide to what extent they can interpret or equate facets of the model with the reality they are trying to model.

Their own personal preexisting worldviews will influence this essentially creative process.

So I would say that there is no hard and fast rule; if there is a simple logical equivalence there shouldn't be a debate.

Unfortunately such models are few. Infinities of one sort or another are a common though not the only source of difficulty.

I doubt very much that problematic cases can be reduced to a divergence at any particular point, that sounds like a mathematicians attempt to define the problem.

A healthy dose of physical insight not to mention experiments if possible are better ways to test models.

If mathematics is discovered, this implies that the objects in question preexisted and existed before anyone was there to have thought about them. Believing such a thing seems like an act of faith (or perhaps confusing the model with reality).

Ian,

There are, in my mind, two kinds of maths; the abstract one and the real one.

The abstract maths goes as follow. I have 1 dollar in my pocket and 1 dollar in the bank. I have 2 dollars total.

The real maths, on the other hand, is pretty much how we first learn about it in school. We count objects by gathering them together. The actual addition consists in bringing closer together the items added. Similarly, planets gain weight by actual aggregation of matter, not by banking on the locally available matter. As we can see, the real "metaphysical" addition carries a geometric component while the abstract addition does not. The metaphysical addition is never "completed" because it comes down to close juxtaposition. For example, 1 + 1 becomes (1+1) but never two or one because they are real entities, they never completely melt into an undefined mush total (one). It would seem that the fact of existence is a discrete very small scale process. their effect add up to as more mass but their discreteness remains. (But their number will vary as in a star collapsed to a neutron star....)

The question is; are our maths truly metaphysical in that they represent the real universe or, are they just abstract representations of our perceptual reality? Physics being empirical, I would think the latter is the case.

Marcel,

Ian,

Allow me to explain...

Science is empirical. What does it mean? It means that we recognize not knowing about the underlying reality. It means that we accept this ignorance because we have found about 300 years ago a pragmatic approach to this situation. We simply treat this universe as a black box. We ignore the content of the box and concentrate our study on our interaction or experience (empirical) with the box. By studying our experiences with the box we have come up with regularities and some possible image and idea of what the box contains. These are our laws of physics and the models that we can infer from them. But no matters how pointed our empirical method is, no matters how sharp and detailed our models are, they are still modeled and framed on the requirements of proof within the empirical system. In other words, the empirical method was meant to study our experience of the box, never to find its content, which must be addressed in a metaphysical approach. No matter how wonderful our science may appear, it is just child's play. Without knowing the content of the box, we do not have any idea of what we are really doing. This is the limit of physics. We don't do or understand as much as we could and should. The content of the box is about the two following metaphysical questions; what is the universe made of and what makes it evolve by itself? The two pillars of metaphysics: substance and cause.

Somehow on the way, we forgot half of the question. This ignorance is in fact an oversight.... Your maths pertain to our experience of the box, not to its content.

Marcel,

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

Dear Ian Durham,

My two messages were meant to test the water. It seemed most likely that you might be interested in comments that are based upon the acceptance of most of theoretical physics as representing reality. In that case, I would not pursue a discussion based upon the possibility that theoretical physics might have gone astray right from its beginning. Actually, everything I bring up in the blogs, forums, and contests are ideas for which I have already developed the specifics. My essay in the first contest put some of those specifics forward.

That essay was a limited, out of context, test to see if there was any interest by others in the possibility that theoretical physics went astray right from the start. I work alone and place my work on the Internet at my own website, because, that has been the only way available to proceed. I have written a great deal, with specifics, about everything I mention to others. I just don't bother bothering others. Others are either interested or they are not.

I will post a sample of my view of how the mathematical representation of theoretical physics diverged early from reality. Please say whatever you honestly feel about it. This time I will show how I think theory might have been started without introducing indefinable theoretical properties. It will be a simple, but also seriously different perspective. If you deem it to be unhelpful, then I will drop it. It will be my next message.

James

"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