Readers' Choice: True Lies: Why Mathematics is an Illusion

This statement:"According to quantum mechanics, you cannot ask what the properties of a quantum particle are before it is measured. Prior to observation, the particle exists in a superposition of multiple contradictory states."

There's no simple yes/no or true/false answer to questions about the state of the particle." is a puzzling statement?..isn't a "measure" actually allready asking the !question" ? To measure something is to ask and thus locate a particle's properties? Being measured is axtually a default knowing "before" question?

Reducing observation down to simplistic "yes/no" questions is not a viable process, as stated in the article, but what if we looked at a reverse perspective?..for instance, any macro entity such as an observer, when trying to locate a micro entity/paticle is actually looking from large-scale to small-scale, so what about the micro particle looking outwards..any quantum actually "know" it is being measured by default, a quantum will "know" about a MACRO entity long before the entity has located the quantum!

If one tries to locate a needle in a haystack, one could find it by chance quite quickly, or it may take an eternity, something large trying to locate something small, by its very nature has a lot of variables. Now reverse the "logic" process, what if the needle could relay its knowledge of the "seeker2, the macro entity trying to locate it, if it could signal to the observer, here I am!, look over here!...then there would be interesting alterations to the Macro to Quantum, information exchange domains?

The very fact that a Macro observer moves the Quantum needle from location to location, a brownian motion like effect by default? The needle is no observer, it has no knowledge of hide'n'seek, even though it is being moved by the act of observer trying to locate it, the observation is only one-way.

Now logical sense may have limitations, by the process of cross scale domains?..would it make sense to walk into a haysack barn and proclaim:Needle, Iam not looking for you... the questioning seems to have relevance?

Who can doubt that actually looking for an actual sowing needle for instance, causes the sowing needle to be NOT located with ease, yet as soon as one gives up and turns to walk away, the needle appears under your foot..OOWWCHH!

It is not that trees in a forest do not exist or fall over when nobody is measuring them, its the facat the quastioning itself, appears to constrain the actual outcome of probable answers"

I do really admire the ones who take the challenge to unravel the cpmplex observer/observation_measured/measure fields, quite interesting.

Lawrence,

Hopefully you will agree that Leshan, Döring, Paul Valetta and someone whose essay is characterized by frequent use of "in some sense but" might simply be too old as to follow the logics of Euclid and any critically thinking child: Point, line, etc. are different from reality in that being infinitely small or thin.

I consider the idea that points are like pebbles or like symmetrically arranged patterns on a dice a cancer to be removed from mathematics. Let's admire the wisdom of Alexandria which was summarized by Euclid: A point is what does not have parts.

The usual notion of a cut means to separate something by placing a knife between parts. Dedekind's cut is not in position to separate two continua from each other. Accordingly we have to re-understand the notion singularity and should be very skeptical concerning claimed singularities in physics. This does not mean they are useless. I enjoy using them as I am using points and lines, which strictly speaking do not exist either.

I see children, very young physicists, and to some extent mathematically proficient engineers like Wolfgang Mueckenheim who is a dean in Augsburg and a professor who lectures mathematics best suited to contribute to an overdue correction of some basics of mathematics. Do not get me wrong. I do not entirely share his views. Old chairs shy back from the huge heap of rubble already feared by Fraenkel.

Regards,

Eckard

Lawrence,

When you reminded of controversy on set theory you certainly meant how Cantor managed to get published with support by Weierstrass and Hurwitz and despite of Kronecker's rejection. His populist attitude was widely appreciated: "The essence of mathematics is just its freedom." While there was not really a need for a set theory, Cantor's stunning proofs were seemingly correct. Kronecker gave up and died. In 1884, Cantor got insane for the first time because he failed to provide an already announced proof of his CH. Hilbert declared in 1900 the CH a most important problem. In this case and also in argumentations by Zermelo and by Fraenkel, sets of points were not at all put in question. From the very beginning in 1872 Cantor agreed with Dedekind and perhaps virtually all other mathematicians on the fallacious idea that there must be more real numbers as compared to the rational ones because the latter are a subset of the former.

My knowledge of the controversy is mainly based on original papers and Fraenkel's 1923 book. Christian Betsch who earned a 1,000,000.00 Deutschmark price, which was a fortune after inflation, also did not deal with the question whether numbers are points or measures.

Having looked in vain for any possibility to justify Cantor's naive cardinality, I see Galilei's and Euclid's insights still valid and a huge crowd of mediocre mathematicians busy with a tempest in a teapot for more than a hundred years.

The late Fraenkel cautiously admitted that set theory merely made mathematics more "interesting". I would rather say more bizarre.

Let me stress that I am not an antisemit. Cantor's opponent Kronecker was a wealthy Jew while Cantor himself got a Catholic education from his mother. Stupid Germans said one has to be Jewish as to understand the set theory. Wouldn't this possibly mean that Jews can in general be mislead more easily? The list of Hurwitz, Hadamard, Bernstein, Mittag, Schloemilch, Hessenberg, Hausdorff, v. Neumann, Bernays, Goedel, Fraenkel, Grothendieck, Robinson, Cohen, Levi, etc. is not less impressive than the list of those who Cantor himself called opponents of his transfinite numbers including Cauchy, Galilei, Gauss, Hegel, v. Helmholtz, Kant, Kronecker, Leibniz, and Newton.

Fraenkel himself stated that there were only a very few who resisted: Poincaré, Brouwer, and I would like to add Kolmogoroff, Heyting, Lorenzen, Bishop.

However, science is not a democracy, and it must not be attributed to belief. What ultimately counts are only solid arguments, experimental verification and valuable applications in practice.

Regards,

Eckard

Singularities in physics usually indicate a breakdown of a a physical principle. Singularities in general relativity tell us some more global principle is involved, such as quantization.

One problem I have with commenting on set theory is that I am not sufficiently educated in the topic to make much contribution to any such discussion. Mathematics builds up model systems, and one can build systems which avoid some of these curious issues of seemingly paradoxical cardinalities. The mathematicians do though like to work with the most general or powerful (powerful in being able to prove the widest array of theorems) system possible. We have a similar situation in physics with string theory, which makes a vast array of "predictions" that are not all realistic, but which is able to address certain physics problems better than the alternatives. ZFC theory does seem to fit the bill for mathematicians who work on these issues.

Cheers LC

Inevitably it seems, when physicists get stuck for an answer, the fault must lie in the mathematics. To me, that is sort of like blaming one's inability to tell a story on one's lack of vocabulary--and then blaming the audience for not understanding one's newly-invented terms when one decides to tell the story anyway.

Even in the days before mathematics and physics were truly separate disciplines, new mathematical methods came under suspicion by those seeking more intuitive, even "obvious" and simpler ways of describing perceived reality. The application of zero as a number, complex analysis, Newton's fluxions, Riemannian geometry ... examples abound for mathematical theories, methods and techniques that resisted physical application until forced by necessity.

One needn't worry about replacing mathematics with another system. It will replace itself. The most social and adaptable of the arts, mathematical language evolves just as natural language evolves--to meet the demands of describing new experiences. One need be reminded that almost all that we know of objective reality, however, is counterintuitive, not simple nor obvious until incorporated fully into the culture. And why should we be concerned about closing the gap between language and meaning? That gap is where understanding lives, the place where we find objective knowledge, and where we find the most interesting questions to ask of the "subtle, but not malicious" reality that we share.

Tom

Tom and Lawrence,

Given, my - as I am claiming - not entirely unsubstantiated suspicion that SUSY is based on a mistake will not be refuted by experiments with LHC or the like. Might I then hope for contributing to a more humble readiness to put at least the worst and not even positivist speculations in question? Zeh (4th ed. , p. 130) admits that there are reasons why white holes are unrealistic. I doubt that past, future, naked and other physical singularities must be taken as reality.

I looked for the origin and early meaning of the notion singularity at the time of Bernoulli and Euler and found it closely related to what I suspect. Let me explain it in the language of a boy who knows a ball and the globe. Is the middle point something special? No. It just belongs to the ideal equal distance from surface.

Triviality stems from three ways. The model has a node of ramification. Two streets in reality do not.

Seemingly, a theoretical physicist should reach the age of Methusalem as to have a chance for getting educated enough for answering my questions in a manner that does nobody understand. I see already Lawrence overqualified in mathematics.

Again, I consider my reproach worth: It is more reasonable to consider numbers as measures and not as points. I just found the notorious lack of clarity also in a conference paper by Jeffrey, Labahn, et al. of Wateloo on integration of signum, piecewise and related functions. Do we really need true lies, differently defined infinities, differently defined signum functions, etc.?

Indeed, reality and appropriate mathematics are not malicious. At the moment, Wikipedia seems to include a huge collection of mistake-related twists.

I recall that a textbook for engineers showed a function |sign(x)| like this

1 ____ ____ as if it intended to illustrate Lipshitz continuity.

0 V

Regards,

Eckard

Usually in the case of physics the fault of a theory lies not in the mathematics, but in the postulates of theory which result in some obstruction. This occurred with electromagnetism and mechanics, which Einstein first saw when he imagined himself on a reference frame moving with and electromagnetic wave. Similarly Planck realized that a problem with getting the spectra of a black body right was due to the assumption of having harmonic oscillators with a continuum of energy. By removing this obstruction then the apparent paradox is removed. From a mathematical perspective this does usually mean the adoption of a more general or powerful mathematical system for theoretical formulation.

When it comes to set theory that is pretty far removed from physics by and large. It is sometimes called metamathematics, for it concerns itself with the foundations of mathematical thought and how mathematical models are consistent. As I indicated I have really a tangential knowledge of this subject, so I am probably the last person to work up a more general foundation which could replace ZFC, or contribute much in such an effort.

This is not directly related to the issue of Topos theory, which is a categorical system of sheaves. There the type of algebraic geometry one works with is dependent upon the category (functor system etc) employed. This gives rise to this idea that mathematics in certain setting can "Lie."

Cheers LC

Yes, usually the fault lies not in the mathematics, and I would like to add that the words like theory, theorem, theology, theocracy, Theodor, etc. relate to theo=deo=deus=Zeus=god.

Maybe, for this or a similar reason Euler did not call his work a theory of ships?

You Lawrence uttered the common opinion:If there are obstructions, then the mathematics must be made more general. I strongly object in cases where I consider a theory already as too general.

Is it justified to use future time for an analysis of what relates to a process that is finished? No.

Does it make sense to allow for a diversity of arbitrary to choose definitions e.g. in connection with a function at a discontinuity and its value(s) there?

No.

Does the complex representation of a function that has measurable values only for positive arguments really provide an additional degree of freedom? No.

Isn't it a at least extremely risky to speculatively interpret without justification as real not just the obviously matching solution of an equation that fits to some extent to the physical reality but all of it, e.g. all Schwarzschild solutions?

I consider Gantor's utterly naive theory an exception, something really wrong and in the word ueberabzaehlbar reminding of Nietzsche's Uebermensch. Cantor admitted having got it directly from god.

So far I do not yet understand why Euclid's correct attribution of a numbers to the measure of a pieces of a line was forgotten. Maybe, the Romans were in mathematics the true "Vandals".

Regards,

Eckard

As a layman, I am surprised:

http://anubis.dkuug.dk/JTC1/SC2/WG2/docs/n2708.pdf

Let me explain why I consider it relevant for the relationship between mathematics including topos theory and physics that some Greeks did already operate with zero.

Most likely those like Platon (427-348) and Pythagoras (570-500) are to be blamed for the attitude of those mathematicians not to accept zero as a number,

negative numbers, etc., who expected that the world is based on laws of mathematics and anything is number.

I first looked at Descartes for several reasons:

- He referred to the rigorous work by Euclid.

- He used the understanding of numbers like points on a line.

- He prepared the invention of calculus by Fermat, Newton, and Leibniz.

The sloppy understanding of numbers like points on a line was, however, perhaps ubiquitous already before Descartes. Albert of Saxony, a rector of the university of Paris, founder of the university of Vienna and finally Bishop of Halberstadt was influenced by Buridan and Ockham. He compiled the knowledge of his time. He wrote: "There is no simple concept of a point, a vacuum, or the infinite, and although imaginary hypotheses provide an interesting detour, physics must in the end provide an account of the natural order of things."

Did he not know Euclid's uncorrectable definition of a point?

He wrote: "If A smaller B then there exists a quantity C such that A smaller B smaller C."

I suspect that the strive for mathematical strength was not agreeable with philosophical points of view as mentioned above. It was in this sense a strength of the middle ages to loose rigorosity.

Dedekind's attitude was then also opposite to Plato's. We may better understand the history of mathematics and physics when we are focusing on the question how the scientists tried to cope with unavoidable pragmatism and the need to accept physical restrictions to the much wider mathematical possibilities. This should give rise for a scrutiny of a large part of modern, in the sense of utterly speculative, theories.

Eckard

Correction:

A smaller C smaller B.

Having dealt a little with ancient and medieval mathematics, I changed my attitude towards Dedekind. His somewhat bewildering claim to create a new number was indeed a brave antithesis to Kronecker who considered the natural numbers as made by god.

Medieval mathematicians often equated the number one with god. "I am the Lord your God. You must not have other gods beside me."

The rational numbers are based on the primary measure 1. Irrational numbers may also be considered numbers if one admits different measures as the basis, e.g. the diameter of a circle or the diagonal of a square.

What about Cantor's idea to count in excess of infinity, I maintain my objections. Cardinalities like aleph_2 etc. are nonsense. While any irrational number is not exactly addressable with reference to the basic measure 1, there are infinitely many "gods" in the sense of - as Peirce understood the mere possibility of - infinitely many basic measures that would allow their own system of counting. The genuine continuum could be imagined to consist of all of them together. One must, however, not hope for unification of all "gods" in a manner other than calculation as if the reals were rationals. It is reasonable and sometimes necessary to accept that continuity and the measure-one based numbers exclude and mutually complement each other.

Eckard

truth of the universe is not in numbers but in proper understending of time

yours amritAttachment #1: Linear_Time_PastPresentFutuure_is_based_on_Neuronal_Dynamics_Physical_Time_is_based_on_Motion.pdf

There is a whole lot of machinery involving n-tupes of points on a manifold or the real line. In particular metric geometry is founded on this. The role of measures come with the integration of functions over the reals, or some manifold in L^p spaces. This is particularly of importantce when the function has some strange support, such as "continuous almost everywhere." I don't think that the point on a space perspective is inconsistent with the concept of a measure.

Cheers LC

I did not yet manage reading Euclid`s book VII with his definition of a number. So I only guess from secondary literature that already he go it correctly: The primary basis of counting is the number one. It denotes an extension, in particular a length, i.e. the distance between two points. Countable are repetitions and splits of it also in combination, i.e. one-based numbers that we may obtain with the four basic operations addition, multiplication, division, and subtraction. Seemingly these measures are not much different from points because we tacitly assume that they all refer to the point zero, called the neutral element of addition and the point one, the neutral element of multiplication. As there is no limit to the amount of countable, i.e. rational points, there is also no limit to the amount of countable measures. Accordingly, there are uncountable, i.e. irrational measures.

What justifies my obstinacy? While the addition of dimensionless points does not yield for instance a one-dimensional line, measures do have a priori the dimension of their basic element. We need not the unrealistic fiction "all of infinitely many" for getting a different quality by resorting to an unimaginably large quantity. Infinity was originally and is according to all reasonable logic still simply the property to have no limit. It is not a quantity even if god-related speculations led to absurd rather than merely abstruse twists.

Is this arguing perhaps correct but pointless? No. The reason for me to deal with this topic emerged from practice. I am still not yet in position to overlook all consequences. The point on a space concept follows with some caveats from the measure concept. EPR and the question whether or not a single point does exist in a continuum prove pointless. It does not matter that Buridan's ass seems to be first mentioned not by Buridan himself but by Pierre Bale (1647-1706). It nicely describes the problem. If 0 is a measure then I consider |sign(0)|=1 correct. The constructivists/intuitionists suffered from troublesome attempts to remedy the consequences of basing mathematics on sets of points.

Strange constructs like Weierstrass's monster and Cantor's dust were interesting from mathematical playground point of view. Did they have implications in physics?

Eckard

I can only say that point-set topology used to underlie differential geometry and algebraic geometry has yet to find successful rivals. I have not studied the history of this in more recent times, but there have been attempts to construct alternatives in the 20th century, but these seemed to lead to greater difficulties or are not sufficiently powerful to build rich mathematical structures.

The Wierstrass monster set is used in electrical engineering. You might have noticed how cell phones don't have an antenna sticking out. The problem back then was how to get an antenna that received on a wide bandwidth. The solution was to construct a chip with an epitaxial form of the monster set. This permitted a wide range of EM reception, and the antenna sticking out of the phone is removed. As for Cantor dust, that is what an energy surface or torus of dynamics turns into when "punctured." So the Cantor dust is something which is a backbone of chaotic dynamics.

Cheers LC

Lawrence,

Weierstrass was not recognized until he shocked the mathematic community with a monster function that was already found by the theologian Bolzano. Weierstrass claimed that this function is everywhere continuous and nowhere differentiable. To my knowledge, he did not explain why: This "pathological" behavior is bound to a fictitious completeness: The function is a sum of an actually infinite amount of terms. Even the best approximation with as many terms as you like behaves as expected.

You mentioned fractal antennas as an application. They are not only rather dissimilar to Weierstrass's monster but they are fundamentally different from it as is anything that belongs to reality. It is impossible to build antennas that do not have an upper limit of frequency. That's why I consider the belief elusive that Weierstrass' monster, some admittedly nice work using infinity by Reuben Goodstein, and the like not immediately relevant for reality. Incidentally, already the simple circle and all belonging harmonic functions that are thought ranging from minus infinity to plus infinity do not exactly describe something in reality.

I fear, you will not continue discussion because I and you cannot convince each other. Let me confirm your argument that alternatives to set theory based on points were not successful so far.

I would appreciate any hint to some attempt to resume what perhaps already did Euclid: Consider the notion of number based on the primary measure one, i.e. counting with respect to the distance between two points one of which is the neutral point of addition/subtraction, the other one is the neutral point of multiplication/division. I am pretty sure that you thought of the few constructivists, in particular Brouwer, Weyl, Heyting, and Kolgomoroff. To my knowledge, they tried to create an alternative set theory also based on points. The only ones who were close to my suggestion seem to be Baire, Borel and Lebesgue.

Constructivism was perhaps prompted by some unrealistic aspects of formalism while it proved obviously less efficient. I hope, choosing the measure as the primary notion of number will lead to a more satisfactory result: both realistic and efficient. After I will have found Euclid's notion of a number in book IIV, I would call it neo-Euclidean. Cantor's dust has measure zero, regardless whether or not a close approximation might be valuable.

Attribution of always just one number y to a number x is anyway no must for engineers, who are familiar with hysteresis. Purification of mathematics from illusions will be a good basis for the purification of physics from much more phantasm.

Regards,

Eckard

Dear Everyone,

You have to see this paper:

http://www.scribd.com/doc/24674634/The-Sixth-Problem

Of course the application of the monster group as a fractal antenna must have a cut off. so it is a finite element approximation. Similarly numerical representations of fractal cut off the iterations as well. More later.

Cheers LC