The limits of mathematics

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.

Dear All,

Mathematical representation of absolute truth, zero = i = infinity can be deduced as follows

If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true

If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true

If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true

If 0 x i = 0 is true, then 0 / 0 = i is also true

If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true

It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero to infinity. We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".

This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.

Love,

Sridattadev.

This is an elementary slapdash answer before I lose my cool and run away from this conversation.

You state above: "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. Now, this obviously does not mean any instance of infinity since, as an example, infinity is necessary to resolve such physical paradoxes as those conjectured by Zeno. But there *are* instances where mathematics in some infinite limit ends up describing something that is not physically possible. There are probably plenty of examples that don't involve infinity."

My intuition and mathematical researches lead me to feel strongly that insight into the above problem can be gained by a fuller understanding of the relationships between Thought, Mathematics, and Physical reality. This can be accomplished by comparing the math describing both thought and reality. The mathematics (symbology) is the same in each case, even though the two Subjects: Thought and Reality are different Kinds of subjects. So the math contains the clues.

Two specifics come to mind: 1)George Boole's law of thought: X^2=X, and 2)the Quantum Mechanical linear operator |Y>

Dang, I had a totally brilliant thread there for you to contemplate, and then deleted most of it with an accidental keystroke. But because of the lack of feedback I get in these threads, that should be no problem for anyone. So, I ain't worth retyping. Here I'll just finish the last cut off sentence above:

2) the quantum mechanical linear operator |Y>

WTF! Warning, do not attempt to type the linear operator |Y> .

[|Y>

The square of the product of a ket and a bra vector (with the ket on the left of the bra) is equal to the original bra and ket. This is impossible to represent in this thread symbollically, apparently, as it deletes all. Anyway, this is the same form of equation as in the Boolean law of thought.

See my essay for symbolical demonstration, and how the interpretation of these equations is accomplished. Typing all this a fifth time just ain't gonna happen...

The bra and kets have to be conjugate imaginaries of each other for this equation to hold, of course...

Dear Sir,

Number is a property of all substances by which we differentiate between similars. If there are no similars, it is one. If there are similars, the number is many. Depending upon the sequence of perception of "one's", many can be 2, 3, 4...n etc. Mathematics is accumulation and reduction of similars, i.e., numbers of the same class of objects (like atomic number or mass number), which describes the changes in the physical phenomena or object when the numbers of any of the parameters are changed.

Mathematics is related to the result of measurement. Measurement is a conscious process of comparison between two similar quantities, one of which is called the scaling constant (unit). The cognition part induces the action leading to comparison, the reaction of which is again cognized as information. There is a threshold limit for such cognition. Hence Nature is mathematical in some perceptible ways. This has been proved by the German physiologist Mr. Ernst Heinrich Weber, who measured human response to various physical stimuli. Carrying out experiments with lifting increasing weights, he devised the formula: ds = k (dW / W), where ds is the threshold increase in response (the smallest increase still discernible), dW the corresponding increase in weight, W the weight already present and k the proportionality constant. This has been developed as the Weber-Fechner law. This shows that the conscious response follows a somewhat logarithmic law. This has been successfully applied to a wide range of physiological responses.

Measurement is not the action of putting a scale to a rod, which is a mechanical action. Measurement is a conscious process of reaching an inference based on the action of comparison of something with an appropriate unit at "here-now". The readings of a particular aspect, which indicate a specific state of the object at a designated instant, (out of an infinite set of temporally evolving states), is frozen for use at other times and is known as the "result of measurement". The states relating to that aspect at all "other times", which cannot be measured; hence remain unknown, are clubbed together and are collectively referred to as the "superposition of states" (we call it adhyaasa). This concept has not only been misunderstood, but also unnecessarily glamorized and made incomprehensible in the "undead" Schrödinger's cat and other examples. The normal time evolution of the cat (its existential aspect) and the effect of its exposure to poisonous gas (the operational aspect) are two different unrelated aspects of its history. Yet these unrelated aspects have been coupled to bring in a state of coupled-superposition (we call it aadhyaasika taadaatmya), which is mathematically, physically and conceptually void.

Mathematics is related to accumulation and reduction of numbers. Since measurements are comparison between similar quantities, mathematics is possible only between similars (linear) or partly similars (non-linear) but never between the dissimilars. We cannot add or multiply 3 protons and 3 neutrons. They can be added only by taking their common property of mass to give mass number. These accumulation and reduction of numbers are expressed as the result of measurement after comparison with a scaling constant (standard unit) having similar characteristics (such as length compared with unit length, area with unit area, volume with unit volume, density with unit density, interval with unit interval, etc). The results of measurements are always pure numbers, i.e., scalar quantities, because the dimensions of the scaling constants are same for both the measuring device and the object being measured and measurement is only the operation of scaling up or down the unit for an appropriate number of times. Thus, mathematics explains only "how much" one quantity accumulates or reduces in an interaction involving similar or partly similar quantities and not "what", "why", "when", "where", or "with whom" about the objects involved in such interactions. These are the subject matters of physics. We will show repeatedly that in modern physics there is a mismatch and mix-up between the data, the mathematics and the physical theory.

Quantum physics implied that physical quantities usually have no values until they are observed. Therefore, the observer must be intrinsically involved in the physics being observed. This has been wrongly interpreted to mean that there might be no real world in the absence of an observer! When we measure a particular quantity, we come up with a specific value. This value is "known" only after the conscious or sentient content is added to the measurement. Thus, it is reasonable to believe that when we do not measure or perceive, we do not "know" the value - there is no operation of the conscious or sentient content is inert - and not that the quantity does not have any existential value. Here the failure of the physicists to find the correct "mathematics" to support their "theory" has been put forth as a pretext for denying reality. Mathematics is an expression of Nature, not its sole language. Though observer has a central role in Quantum theories, its true nature and mechanism has eluded the scientists. There cannot be an equation to describe the observer, the glory of the rising sun, the grandeur of the towering mountain, the numbing expanse of the night sky, the enchanting fragrance of the wild flower or the endearing smile on the lips of the beloved. It is not the same as any physical or chemical reaction or curvature of lips.

Mathematics is often manipulated to spread the cult of incomprehensibility. The electroweak theory is extremely speculative and uses questionable mathematics as a cover for opacity to predict an elusive Higg's mechanism. Yet, tens of millions of meaningless papers have been read out in millions of seminars world wide based on such unverified myth for a half century and more wasting enormous amounts of resources that could otherwise have been used to make the Earth a better place to live. The physicists use data from the excellent work done by experimental scientists to develop theories based on reverse calculation to match the result. It is nothing but politics of physics - claim credit for bringing in water in the river when it rains. Experiment without the backing of theory is blind. It can lead to disaster. Rain also brings floods. Experiments guided by economic and military considerations have brought havoc to our lives.

Regards,

basudeba

One real Universe is eternally occurring in one real here appearing moving in one real infinite dimension at one real universal "speed" of light, once. One real Universe can only be accurately describing by the abstract postulated symbol 1, once. Newton was wrong when he associated abstract planetary and astral motion with abstract states of inertia. Einstein was incorrect when, after proving that no real physical state of inertia was possible, he nevertheless asserted that planetary and astral motion could be accurately described by measurable states of relativity. One real Universe can only ever be eternally appearing in one real state, once. Everything moves at the one Universal constant "speed" of light once. I know that scientists claim to have experimented with laser beams and motion detectors and have proved that a fabricated light can move through a vacuum tube at approximately 186,000 mps, but they never took into consideration the fact that the laboratory they conducted their experiments in was travelling at the Universal constant "speed" of light at the time of the experiments as were the scientists who conducted the experiments. There is no such a thing as mathematics. Reality does not have a plurality of anything, real or imagined.