The limits of mathematics

I just found your site. I am in love. I just forwarded the link to all I know. Good luck on future growth. I look forward to it!!

Ian,

I was involved in a discussion at another website. My involvment was not as a main participant. I have copied edited parts of my few messages and include them below. I have removed all texts except for mine. The point I was making is that thermodynamic entropy is not yet explained. Skipping past it to expressions of statistical mechanics is not satisfying to me. I think the mathematical expression defining thermodynamic entropy is an example of the Limits of Mathematics. I began my posts with:

"Saying that thermodynamic entropy is energy in transit divided by temperature is not, I think, an answer to: What is thermodynamic entropy? What did Clausius discover? Whatever it is, it requires the passage of time. Clausius allowed for absorbtion of energy under conditions of equilibrium. Statistical expressions do not include this dependence upon time."

"The T in the definition of thermodynamic entropy represents thermal equilibrium conditions. There is no fluctuation in temperature included. Yet energy in transit is included in the Q. Energy is absorbed over time under conditions of thermal equilibrium. I am not arguing that this theoretical ideal condition is possible in the real world. I am saying that Clausius did discover something of fundamental importance and it is not yet explained to this day."

"Since the derivation includes only long established macroscopic type properties, though they relate to the internal state, shouldn't it be expected that thermodynamic entropy should be explainable as a similar macroscopic property."

I was then referred to this Link to Paper for a more logical and axiomatic explanation of thermodynamic entropy. I read the paper and responded:

"I see the paper as being analogous to reverse engineering. Equilibrium and entropy are assumed as givens. When the authors assume conditions of equilibrium, then they have already introduced temperature into their analysis. It is there from the beginning. It is represented by the presence of equilibrium. The scale used to quantify temperature is introduced later, but the scale is not the property of temperature. Equilibrium is the condition of constant temperature. Different conditions of equilibrium represent different temperatures. They represent conditions of relative hot and cold. The practice of avoiding using these words and the mathematical symbol of T for temperature is not sufficient for saying that they are absent in the analysis until derived later. Changes in levels of equilibrium, no matter how accomplished, introduce the flow of energy as a given. The analysis is formal and axiomatic, very logical. I prefer relying upon the original measureable properties. Entropy is not explained in a mechanical sense. It is not described as a measurable property analogous to heat, temperature, pressure or volume.

I still expect that it should be explanable as a macroscopic classical style property. I think that its mathematical expression does contain a contradiction. Heat is energy in transit while equilibrium appears to exclude energy in transit. I think that this contradiction, or apparent contradiction, is what needs to be resolved, and, thermodynamic entropy will then be revealed. Another way of looking at this is to say that: When we finally understand what temperature is, then we will quickly understand what thermodynamic entropy is."

I had stated earlier that: "Temperature is an indefinable property with indefinable units of measurement." My point is that temperature is not yet explained.

There was no interest shown or discussion regarding my last message. Anyway, I post it here not because it represents anyone else's view but my own, rather, I submit it for consideration, and possible correction by you and others, as a mathematical solution without a clear physical interpretation, and, a possible classic example of the limits of mathematics.

James

Ian where are you, and why you don't speak to me ...

Ian:

I like the idea of attempting to identify the limits of mathematics because insight into this issue would help to flag the moments when physics or other subjects, such as economics, might be going off the deep end. It is a profoundly challenging question, though, and it sits at the intersection of philosophy, mathematics and science. This is one of those situations where insight into a large question is gained by asking an even larger question.

So let's ask instead: What is the difference between abstract mathematics - with no applications - and real mathematics? A good argument can be raised that there is no difference for one simple reason - all mathematics is abstract. There is a profound difference between useful and real. Abstractions from reality can be exceedingly useful, but they aren't real - they are idealized mental forms that are intentional simplifications of reality. The number 3 seems very real, and so does pi, but there is a reason why you can't see either number written in the sky or on the waves of the sea - these numbers exist only in your mind. They spring from the reality that defines them and makes them meaningful in more than one situation. We shouldn't be surprised that they are useful in multiple situations - we chose them as useful abstractions from reality for exactly that reason.

Long ago, I browsed through a book on the Tao in a library, and didn't find much useful, but one idea resonated strongly with me. It was stated something like - the abstraction of the thing is not the thing itself. This is a powerful and useful observation. It turns out that the Tao has many translations and I have never again seen this quote in exactly the same way, but the idea is eternal. The word cow is not a cow, A picture of a cow is not a cow. Your thought of a cow when you see the word cow or smell cows is not a cow. Our understanding that a cow herd contains individual cows does not create any cows. Only that individual creature on Farmer Jones' farm, named Bessy, calmly chewing grass with a bell around her neck as we watch her with our very eyes is a cow. She is unlike all other cows - a reality we forget when we abstract her.

For every cow there are many abstractions of it. Likewise, it is possible to abstract reality into mathematics in a infinite number of ways. But if our goal is to understand reality we should never confuse it with mathematics or with any of our abstractions. Physics and other sciences begin to diverge from reality the instant this confusion is made.

Stan

Dear Stan,

Well, applied mathematics is applied abstraction. Someone said: "The map is not the territory".

Hopefully, the community of physicists will not feel too offended if I reiterate that predominant physics is ignorant of something that is well understood by common sense: abstracted time is different from real time.

In particular, W. Ritz was correct in 1909 when he insisted that future events cannot influence the past while Einstein preferred to stick on the traditional belief in an a priori given time from minus infinity to plus infinity. Ritz died soon.

Maybe, Minkowski got ill and died because he felt being wrong with his exciting idea of a belonging complex spacetime.

If I recall correctly, Schwarzschild died in 1916 before his complete solution for the metric around a point mass was interpreted as reality even for past and future singularity and beyond.

If I compare the about 4 000 Mio € expense for LHC with the 750 000 Mio € agreed in order to finance some almost bankrupt generous European states, I see the LHC worth his money if one will draw the due consequences from the outcome.

Regards,

Eckard

Eckard:

I think Ian senses, along with some other physicists, that physics may have gotten lost somewhere along the way. He is looking for a method that will flag an abstraction as having no parallel in the real world. There are many ways to approach a problem like this, and if you have an open mind insights can be obtained that give you the ability to tell the difference even when the rule is not absolute.

The solution begins by understanding what abstraction is in general, and then applying that understanding to math. The human brain is designed to abstract our experiences into words so we can form a mental model that we use individually and communicate to others. This ability allows us to continue the process by abstracting some words into numbers or mathematical structures. In other words, we can create abstractions of abstractions, and with math that is how the process begins.

The process of abstraction has no natural limit, so with so-called abstract math we have created abstractions of abstractions of abstractions, etc. This gives us the first probabilistic rule of "real" abstractions. We can state it thus: The higher the level of the abstraction, the less probable an attempted application of the abstraction will work in the real world. This is not the only useful probabilistic rule, but is one of several.

To understand this rule we have to become more concrete by working with examples - a process which is a hint of other rules. Whenever examples are elusive or hard to work with, you have probably landed on an abstraction that is useless in the vast majority of situations. As a first example, consider negative numbers: are they "real"? The answer is, yes they are, but they are more abstract than positive numbers and thus apply to fewer situations. For example, if you are happily crunching away with your equations and come up with a negative human height you immediately know you have an error. Likewise with negative frequencies, negative wavelentghs, negative volume, etc. The artful mathematician might immediately jump in, beat his breast and insist, "I can make people negative 9 feet tall!" That isn't the point. The point is that it is necessary to strain mightily to find the required example, and this is true to the extent that when you end up with a negative human height in a real world problem you almost always have an error in your calculation. What we want to know is the probability of error, given only the nature of the abstraction itself.

The scarcity of negative human height, negative frequency, etc. in the real world should serve as a warning about the danger of over-abstraction. Negative numbers are one of the most useful 3rd-level abstractions (abstractions of abstractions of abstractions), but even they apply to a much smaller set of real world situations than positive numbers. The higher the level of abstraction the more useless the application as measured by the ratio of errors to correct answers in real world problems.

A corollary to this rule is that when you find yourself working with 12th level abstractions, you should have the sense to realize that applications are highly improbable, even if they aren't impossible. You are in a situation where you will make zillions of errors for every useful result, and they will be so subtle you will probably never find them or even sense that they are there.

Whenever you are in this situation, you are a fool if you don't do two things: 1) Seek lower-level reality checks, preferably in abundance. 2) Try to reduce the problem to a lower level of abstraction. If you can't do either one of these things, you have to accept the reality that your model has an unknown but extremely low probability of being correct.

Stan

Ian,

... and why aren't you speaking to me?

Ian:

I think your basic question is first rate. It would be nice if a journal or two dedicated space to this issue because I suspect progress could be made over time. In particular, I think there are probably probabilistic, heuristic, and empirical methods that would be highly useful, even if exact solutions are impossible, which is likely.

In other words, I think this is an important, large question that can't be answered without considerable exploration. If the truth is empirical, this would be an answer some would shun because they are in the habit of seeking pure answers. However, all of physics involves testing our "pure" ontological ideas against empirical observations. An empirical answer is potentially enormously useful.

For example, physics may very well have lapsed into a state where extremely poor bets on abstract ideas are routinely being made because we have no methods, even probabilistic, for gauging when an idea is likely to be absurd on its face.

Stan

Dear Stan,

While I agree with you on a many details, I feel unhappy with your overall attitude. You wrote: "... when you end up with a negative human height in a real world problem you almost always have an error in your calculation." Really? I would rather say you have to carefully check what you lost with abstraction and to find the correct way return. For instance, complex calculus is an excellent tool but do not interpret solutions immediately in complex domain as if they were real.

What about Georg Cantor's attitude "the essence of mathematics is its freedom", I see it a denial of logic constraints and responsibility for serving applications. Your "abstraction of abstraction of abstraction ..." reminds me of Cantor's naive self-made "God's view" (infinitum creatum sive transfinitum) and of Wolframs repeatedly used "in some extent but". I prefer precise judgments. Teachers and judges must not hide themselves behind a value of probability but decide correct or wrong, innocent or guilty. I agree with hopefully many that the probability for some abstruse, allegedly highly abstract theories being inappropriate or simply wrong is rather high.

I blame Heisenberg, Schroedinger, Weyl, and Dirac for their inappropriate reluctance to accept negative frequencies. Even physicists should understand their tools. They obviously didn't.

Did Fraenkel understand the notion point when he justified the obviously naive reasoning by Cantor: "a piece of line densely enough filled with points"? To me infinity is still an abstract ideal, a property, not a huge quantity.

Regards,

Eckard

Eckard:

Primarily what I am saying is that this is a tough question that can't be answered off the cuff. Instead, it is necessary to begin with an information-gathering assessment of what we already know. A close scrutiny of the correspondence between known mathematical objects and real-world situations would be a useful approach that would reveal how the two are related.

Interestingly, this would be an abstract study. The goal would be to identify the features of mathematical objects that reduce the odds they have real-world application. I can't think of any other way to gain insight into this relationship without studying the real cases that we know.

First study the known, then extrapolate to the unknown.

Stan

Stan, To me reductionism up to Zermelo is typically arbitrary, naive and inappropriate. Euclid is still more prudent and convincing to me.

Incidentally, since English is not my mother tongue, I do not understand your sentence: "The goal would be to identify the features of mathematical objects that reduce the odds they have real-world application." Is there a word missing?

My approach is perhaps different. While I am aware of abstractions as precious tools I feel in position to not confuse elements that belong to different levels or domains. I reiterate: Complex domain has its own rules and must not carelessly confused with interpretation of reality. I mentioned negative frequencies. They are a must after transformation of any realistic function of time into complex domain. They are required as a precondition for a correct return to reality.

Alternatively, one may also arrive at physically correct results by avoiding the detour over complex domain. This merely requires to abandon the "God's eye" view of a bilaterally infinite block spacetime.

Cantor's ueberabzaehlbare (literally more than countable) infinite transfinite numbers in excess of infinity reflect his "God's eye" view in contrast to Galilei's interpretation of bijection: "For infinite quantities the relations larger, equal to, and smaller are invalid".

You wrote:"A close scrutiny of the correspondence between known mathematical objects and real-world situations would be a useful approach that would reveal how the two are related." Indeed, aleph_0 simply means never ending, aleph_1 means uncountable, but from aleph_2 on there is no application possible.

"this would be an abstract study" ???

I conclude not just from Georg Cantor's utterance "I see it but I cannot believe it" that he was largely unable for consequent abstract reasoning. Ironically, his abstruse naive point set theory was mystified like too abstract as to be understandable. In that it resembles quantum mechanics, and it relates to EPR.

Cantor persistently denied the 4th logic possibility.

"The goal would be to identify the features of mathematical objects"

Hopefully you are ready for putting the point-set based topology in question and resume a measure based instead.

Eckard

Just as the "logarithmic integral" Li(x) can be used to approximate

how many prime numbers there are under a given number x,

the "non-trivial polygonal number counting function"

which you can find here:

on_polygonal_numbers.pdf

can be used to approximate, (with even greater accuracy)

how many "polygonal numbers of rank greater than 2"

there are under a given number x.

What makes this "counting function" so unusual

(and of possible interest to the physics community)

is that it seems to require the two most important

"physical constants", alpha and mu.

I personally have no theories as to why this should be so,

and am presenting this as a mathematical function only.

I welcome your comments.

Don

Ian,

Thank you for this forum. The linkage between mathematics and physics and its limits is the most important foundational question. My work developing www.zenophysics.com indicates that much of the trouble with physics can be traced to the failure of Newton's calculus to counter Zeno's paradox of motion. Here is the story:

1. Zeno's paradox of motion: If you assume that points are dimensionless and that space is a continuum then Zeno's logic is accurate and motion is "logically" impossible. Take a particle like an electron at a position in space. For it to move it must first move to the point closest to it. The point closest to the electron is at the position of the electron because as a point it is without extent (dimensionless). All the points that the electron must move through are similarly at the position of the electron. There is no motion possible for the electron because the sequence of points that it must proceed through are at the position of the electron, and thus the electron cannot move. The continuum of points is quicksand for motion. QED

Of course something is wrong with the motion paradox. But is the problem with the Mathematical assumptions or the Physical assumptions in the paradox?

2. Can mathematics be changed to remedy this "loophole" that Zeno found? Alfred North Whitehead tried with a "point free" geometry. This concept has not picked up critical mass and the non dimensional point is still the preferred concept. Because of the tenacious of our concept of the point and the continuum I will make them the axiomatic start of physics and say that the start of physics is objects existing at point positions on the continuum of space-time. Math as it stands is basically OK.

3. If the math is OK then something is wrong with the physics in that it allows Zeno's trickery. This trouble was first spotted and remedied by Werner Heisenberg with his discrete quantum jumps which were part of his matrix mechanics formulation for quantum mechanics. Now the electron never has to traverse the points closest to it, it simply jumps over them. Heisenberg overcame Zeno, but he did not overcome Schroedinger and his continuous wave equation representation for quantum mechanics. Heisenberg had it right, reality is that physics has as its start discontinuous phenomena (see zenophysics). Trying to model this phenomena with continuous models will always be flawed.

4. The notion of discrete quantum jumps was put on the shelf. It needs to be taken off the shelf and dusted off and put into service to remedy the trouble with physics. See www.zenophysics.com for how I approach this. I need to proceed from Zeno to the other great enlightened trouble maker Isaac Newton.

5. Newton (and Galileo) flew in the face of Zeno by measuring motion. Motion is real because we can measure it as a velocity that is defined as distance divided by time v=deltaX/deltaT. So far so good.

6. Newton now performs the operation that causes the trouble with physics, he says that by using "logic" we can have velocity at a point and stick it to Zeno. He does this by showing that v=deltaX/deltaT=dx/dt. Every calculus text has a chapter where it is shown that deltaX can be taken to dx in the limit and deltaT can be taken to dt in the limit to produce the concept that a velocity can exist at a point. If the people who look at this chapter are mathematicians they are treated to a valuable exercise in logic. If the people who look at this chapter are physicists they are getting a dose of flapdoodle, because the real world does not work like that.

7. It looks like Newton showed that there is a physical reality to the notion of velocity at a point. He did no such thing. What he did do was split mathematics into two realms the unreal theoretical and the practical. It is in the unreal theoretical math world he defeats Zeno in a realm with no reality. In the world of real physics v=deltaX/deltaT is just fine as long as the deltaX is greater than the wavelength of the object under consideration.

8. An electron at a position in space has no velocity. However, you can calculate a velocity for the electron by measuring a deltaX and a deltaT and making a calculation for velocity.

The methods of calculus have to be rethought and brought up to date in light of the findings of quantum mechanics. This includes the notions of Hamiltonians and Hilbert spaces. Don L. 7/5/2010

Dear Don Limuti,

I also consider the basis of mathematic most fundamental to physics.

If you agree with Euclid that a point has no parts and with Peirce that a each part of continuum has parts then a continuum cannot be resolved in a finite amount of single points and there is no closest point to any point. This is not yet a physical but a mathematical explanation.

What "trouble in physics" did you resolve?

Given I am correct then quantum computing will not work as promised, several paradoxes will vanish, the LHC will not fulfill expectations, and several so far mandatory tenets will turn out at least questionable.

I found out:

1) Pupils of Gauss changed the good old notion of number as a measure to the rather questionable older understanding as a pebble.

2) IR is equivalent to IR and fits better to virtually all original physical quantities with natural zero including volume, area, distance, elapsed time, radius, temperature, pressure, number of electrons and other items ... , probability, ...

3) According to 2) Analysis of measured data does not require a complex valued Fourier transformation but just a real-valued Cosine transformation. The FT is redundant. In addition to the information available with the CT it only provides information on the arbitrarily chosen point of reference.

4) The fathers of QM introduced unilateral functions of frequency directly in a complex domain in a manner they borrowed from common practice where unilateral functions of time are considered the original ones. However they changed to the Hamiltonian point of view. This led to misinterpretation.

5) Perhaps, length contraction does not likewise belong to decreasing and increasing distance.

6) Space-time seems to be ill-founded because it would be anticipatory, and the sign of rotation clockwise or anticlockwise is arbitrary.

7) G. Cantor and his proponents did not refute Galilei's argument: The relations smaller, equal to, and larger are not valid for infinite quantities. As Fraenkel admitted: There is a 4th logical possibility: incomparability, and Cantor's naive definition of a set is untenable.

Regards,

Eckard

Dear Eckard,

The trouble with physics: The axioms of geometry* do not allow quantum particles to move.

I propose to makes the axioms of geometry legitimate for quantum particles by making particles move in a special way. That way of moving is what I call wavelength-hopping. The details are on www.digitalwavetheory.com

Thanks,

Don L.

* 1. Dimensionless Points 2. A continuum of points on a line.

The origin of special relativity has much to do with the origin of dynamics.

Useful axioms or theorems that could act as fundaments for physics exist, but they treat only the static relational structure of physical items and physical fields. These law sets consist for a part of traditional quantum logic as it is defined by Birkhoff, von Neumann, Jauch, Piron and others. Quantum logic says nothing about dynamics and it says nothing about physical fields. The static configuration of fields is treated in the Helmholtz and Hodge theorems. Dynamics couple these static ingredients. When dynamics occurs in steps, then after each step the static status quo is reestablished with new conditions. The laws governing the static status quo are reinstalled as well. In this picture nature bumps in universe wide steps from one static condition to the next one. With other words there exists a global progression parameter. This is not our common notion of (coordinate) time, because that has only local validity. In the new condition the actual configuration of the fields determine what will happen with the influenced physical items in the next step. The physical items are sources (charges) of the fields and when they move they form the currents that cause still other kinds of fields. Investigation of inertia shows that a movement that differs from a uniform movement and a non-uniform distribution of charges or currents cause forces that control the next step. In this way dynamics emerges.

All physical items in universe cooperate in the implementation of inertia. Because electrical charges can compensate each other while masses cannot do that inertia shows up for gravitational fields and not for electromagnetic fields. However, both fields take part in the emergence of dynamics.

The axioms that define the static interrelations between physical items stem from quantum logic. The sketched picture implies that the lattice of propositions about physical systems is controlled by influences that act as a sticky resistance against altering the relational structure of the set of propositions. In nature this sticky resistance is implemented by physical fields. Locally the action of the fields alter the observed properties of the physical items. It is done such that the combination of space and local time gets a Minkowski structure. In this way we get two new notions of time: coordinate time and proper time.

The global progression parameter is not a dimension of a space with Minkowski signature. Thus with this parameter the fields do not encounter a maximum speed of information transfer.

The former statement means that fields are not living in a Minkowski space. How is it then possible that our observable world has a Minkowski signature? It is only possible when during the step a conversion takes place that reshapes the expectation value of the position observable into something that has a Minkowski signature. Something that is capable of this is an infinitesimal transform of a hyper complex number q by a hyper complex number u that is close to unity:

[math]\textit{u}\approx 1+ \Delta \textit{s}[/math]

The transform runs:

[math] \Delta \textit{q} = \textit{u}\cdot \textit{q}/\textit{u} - \textit{q} [/math]

The infinitesimal steps of s and q are imaginary and orthogonal. The rectangular triangle can be closed with an infinitesimal step in the form of the hypotenuse. The corresponding variable is the coordinate time t. Now holds:

[math] \Delta \textit{s}^{2} = \Delta \textit{t}^{2} - \Delta \textit{q}^{2} [/math]

This construct has a Minkowski signature. In order to achieve this, q and u must be quaternions or higher dimensional hyper complex numbers. During the process the real part of q is not changed. It is not participating in the q step. The empty place is filled by the coordinate time t.

If this story is true then the space variable q and the coordinate time t have little in common. They do not form a spacetime construct. However the dynamically changed imaginary part of q forms together with t a Minkowskian spacetime.

Einstein left us his formulas for special relativity, but he never gave an explanation WHY these formulas work. The isomorphism of the lattice of Hilbert subspaces with the lattice of quantum logical propositions suggests that properties of physical items are represented by eigenvalues or as expectation values of normal operators in Hilbert space. Spacetime, with its Minkowski signature does not fit as an eigenvalue of a normal operator. Above an explanation is given for the fact that hyper complex numbers that can act as eigenvalues are capable of converting hyper complex numbers into constructs that indeed have a Minkowski signature.

The transform is indirectly done by a thing called redefiner. In quantum logic it redefines quantum logical propositions and its equivalent in Hilbert space redefines the subspaces that represent both the proposition and the physical item that is described by that proposition. The redefiner does not touch the Hilbert vectors. Thus it does not touch eigenvectors of operators. It represents the action of the Schrödinger picture. In the Heisenberg picture a set of parallel trails of unitary operators cooperate in order to shift the Hilbert subspace that corresponds with the considered physical item. The unitary transforms also shift the eigenvectors of operators. Since Hilbert space acts as an affine space for the change of the representation of the physical item, in both pictures the expectation value of the observed property of the physical item must be the same. In each step the set of unitary transforms cause an infinitesimal transform of the expectation value of q in the form

[math] \Delta \textit{q} = \textit{u}\cdot \textit{q}/\textit{u} - \textit{q} [/math]

as sketched above.

More details can be found in http://www.scitech.nl/English/Science/Ontheoriginofdynamics.pdf.