The limits of mathematics

Hi Constantinos,

i read your explanation of the double-slit-experiment and i have a question to it.

If the emission of an electron (by passing the double-slit) has to be thought as a continous field of energy - consisting of the energy of exactly one electron - and distributed in its strengthness due to the probability-interpretation of QM over the detection-screen in the well known manner of an interference-pattern - then, due to the laws of addition the area exactly in the middle behind the two slits has the highest probability to pop up as a light-spot *firstly*. Therefore the first light-spot after accumulation of enough energy must be in exactly this area. Could this have been really the case in all the experiments? And why are there light-spots in areas where there had to be exactly destructive interference - means, a probability of zero?

Hello again Stefan. I just posted a reply to your other post addressed to me. Thank you for your very good questions. Following up on my reply to your other question, let me just say that there are local conditions on the detection screen (beyond the radiated pattern from the emitted electrons) that we just don't know, making any precise prediction just impossible (even if we were to predict the 'area' where the first flash will occur). We may agree that QM does not 'lie' here. But QM does not provide a 'physical explanation' either. What I am hoping with this is to help provide some 'Physical Realism' to what now has become 'quantum weirdness'.

Dear Eckard,

... your call for experimental confirmation and predictability is well taken and appreciated.

There are some new posts re: double-slit experiment explanation in the blog section of Edge of Physics. Just in case you were not aware of this ...

Best,

Constantinos

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

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