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

Lawrence you said "New physical principles have to be advanced as the primary motivator, which then require some mathematical system of description."

The following all apply to a quaternion energy-space not space-time.

All matter has continuous aforeward change of position along the scalar dimension.(Necessary for gravity and arrow of time.)

All subatomic particles have continuous change of position within quaternion space but move within all spatial dimensions in all directions. (Do not experience gravity and may travel backwards along scalar dimension without causality problems as it is spatio-energetic not time.)

Every energy change is a change in quaternion spatial position and vice versa.

Energy is change in spatial position or the potential to cause change in spatial position (potential energy.)

Every energy change causes another energy change and so every change of position of matter, medium or particle causes another spatial change. There is both conservation of energy and conservation of spatial change.

Space and energy are inextricably interwoven because of the previously stated connection between them. This can be represented by 4 spatio-energetic dimensions in quaternion arrangement. Time (in science) is a measuring tool, using regular change within 3D space (such as decay of an atom or tick of a clock) to measure the observed spatial changes caused by spatio-energetic changes within quaternion energy-space. Time still has to be used as proxy because there can be no direct measurement of 4th dimensional spatio-energetic change.It can be assumed to be regular and continuous for matter (just like time measured within 3D space) although this is an approximation as shown by general relativity.

Because the proposed quaternion energy-space has the same dimensional number and arrangement as the original ideas on relativity that were subsequently mathematically transformed into mathematical space-time, I do not see why the same mathematics can not be used but translated in the interpretation . So the time-like vector is actually a spatio-energetic dimension.I understand that there were objections to another spatial dimension because of the difficulty of locating it. However if it is actually the scalar dimension extending into every object at every point on the surface and through the centre of gravity into afore space then this not only solves that location difficulty but allows gravity to be explained.

All of the basic physics and mathematics could be used with the new interpretation because structurally it has not changed. It is only the interpretation that is different. It just needs translating into non temporal terms when interpretation of meaning takes place, not necessarily during calculation.

I would however like for quaternion mathematics to confirm all basic physics and to exceed its current ability. Quaternion mathematics more closely resembles the structure and behaviour that is necessary to model the universe. A movement along the 4th dimension being a rotation in 3D space. It is non commutative which is also seen within quantum physics. Its wide use within computer modelling already demonstrates its power for modelling 3D structure, giving very natural looking forms.

Re. Doug Sweetster's work, it is interesting to hear your appraisal.I do not think that initial difficulties mean that the whole approach is necessarily wrong though. It is the final result that is most important.

No. Dedekind`s cut does not symmetrically divide the line of numbers into two equal parts. Such division without remaining neutral element is impossible for integer and rational numbers while obviously possible for a continuum every part of which has parts, endlessly. Dedekind explained in §4 "Creation of Irrational Numbers" how his cut CREATES a number. He and Cantor abandoned the old notion of continuum.

Lawrence Crowell wrote: Topos theory does not involve Cantor's transfinite numbers. Topos theory does invoke the Zariski topology, or systems of projective and algebraic varieties which are in general non-Hausdorff. The only connection with set theory is with foundational issues that are purely mathematical. Most people working on this or related issues of sheaf theory in algebraic geometry are not focused particularly on set theoretic underpinnings.

I noticed constructivists who tried a topos without the law of excluded middle.

I see the problem deeper hidden within the limitation of the notion number.

Eckard Blumschein

Eckard Blumschein,

Please forgive the baby steps and baby talk that are necessary for me to take.

Firstly I like the idea of open sets very much because when thinking of particles in continuous motion within quaternion space it is never possible to give a location of a particle as it is forever changing. So it is not really appropriate to talk of distance because of the difficulty identifying a location and because the particle can move freely in 4 dimensions. Distance isn't as straight forward as the line between two points. However if it is squiggling about in 4 dimensional space then that squiggle is an open set. (I was wondering how a squiggle of 4 dimensional change in spatial position could be described.) Keeping track of where the elusive dynamic entity might be in its squiggle also sounds jolly useful.So if I am understanding the idea of sheaves correctly then that is good too.

A problem with the notion of number is interesting. If I use the number 1 to represent a singular object, that number stays with the object. However from one configuration of the contents of space to the next the object is changed. Its position in quaternion space has altered. It has moved along the 4th spatial dimension and this gives rotation within 3D space. All of the subatomic particles within the object will have changed spatial position. Depending upon the object there may have been chemical change within the structure giving a different spatial configuration of the constituent matter. The surrounding medium will have changed in position.

I keep thinking about the animation of the gimbal rings Wikipedia gimbal lock

The point being that 1 is not really just 1. It is a whole sequence of alternative 1s. All may seem to paradoxically exist within 3D space because we do not appreciated that the space and the object are in a continuous process of change with 4 degrees of freedom. So it is not exactly the same 3D space or object. Even though we have attached 1 to the object and it is considered to be the same 1. When an observation is made it is as if the handle of the fruit machine has been pulled. The gimbals rotate with 4 different degrees of freedom and at the precise moment of detection one 1 from the whole range of possibilities is selected as the observed existential reality. Quantum physicists would say the wave function has collapsed.

Giving the identity of 1 to any object or particle is problematic. The only unchangeable thing about object or particle is that it has continuous change of quaternion spatial position and thus possesses energy. That gives it existence.

For a particle its location is changing, direction of spin can change ,charge can change, rotation can change. How then is it truely compatible with identity as a singular thing? A macroscopic object is also continuously changing although this is not generally perceived, except perhaps with regard to biological organisations of matter.

We may designate names for particular stages of organisation such as foetus, baby, toddler, child, teenager, young adult, adult , senior or elder. They are obviously not all the same spatial organisation of matter but may still be regarded as 1. The recognition of spatial difference in organisation could be extended so it is acknowledged that day to day the 1 has changed and is not the same 1 as yesterday or even minute to minute or second to second. That is not how we think about the macroscopic world and the objects we identify as singular and name.

It is therefore perhaps the concept of 1 itself that needs revising to acknowledge that 1 is more that just 1 but many different 1s.

The explanations I have given in my previous 2 posts are compatible with the scientific idea of particles existing as a probability until "wave function collapse" but explains mechanically why that should be so. It unites the probabilistic nature of the sub atomic realm with the apparently definitive nature of the macroscopic realm of everyday experience.

Thinking gimbal rings again. Wave function can then be viewed as description of the alterations in orientation of change of spatial position generated from continuous change with 4 degrees of freedom. (It can be hypothesised that spin is the oscillation or rotation along the 4th dimension as it is not rotation within 3D space.)

Dear Georgina, you say: Giving the identity of 1 to any object or particle is problematic. The only unchangeable thing about object or particle is that it has continuous change of quaternion spatial position and thus possesses energy. That gives it existence.

Yes, "motion" and "energy" are two empirical fundaments of physics. All rest is a description.

yours amrit

Eckard,

Cutting the reals does involve a Dedekind cut. As for cutting the reals into two equal parts, that gets one into the matter of adding infinities. To be honest I leave the matter of axiomatic set theory to people who work on that. A mathematician friend of mine calls set theory "set on your ass theory," jokingly to indicate how the subject has tangential contact with other more realistic mathematics.

Topos theory might be compared to what happens in quantum mechanics. QM does admit the overlap of states which correspond to physically distinct outcomes under a measurement. In topos theory sheaves may have completely different pre-S structures or different algebraic varieites --- elliptic curves, projective structures and so forth. Yet a functor between two sheaves remove these distinctions, so what might be seen as contradictory sets can then obtains as categoriies.

Cheers LC

Lawrence,

while I enjoy "set on your arse theory" I consider Buridan's ass something important in physics. The matter has far reaching consequences and deserves more clarification than I am able to provide this evening. Please find a bit food for thought in topic 527.

If we agree that real numbers are different from rational ones then Georgina is wrong: The distinction between open and closed sets is invalid for really real numbers. oo oo = oo at least according to the really great ones like Galilei.

What about topos theory, shouldn't we consult Grothendieck himself?

Sincerely,

Eckard

Amrit,

thank you for the reply.

The singular thing identified as 1 changes and so its description can change. The 1 thing is not the identical 1 thing from spatial configuration to spatial configuration (moment to moment if using time as proxy). It is possible to notice the change in the sub atomic particle because it is such a minimal "thing" but continuous change is also occurring in the macroscopic realm. However because macroscopic objects are such large collections of particles in complex organisation the subtle but ceaseless change is not readily apparent.

This is the elusive connection between the behaviour observed at the smallest scale with the macroscopic scale. We only perceive macroscopic objects as unchanging and singular in their identity because mostly the change is subtle and therefore overlooked. Occurring on a scale too small to perceive or too slowly to be apparent to our unaided senses. Such as the erosion of a rock. (We also do not perceive the continuous change in quaternion spatial position of all objects, even those considered stationary.) There is a difference between subjective reality of experience that tells us most macroscopic objects are singular and unchanging in identity and objective reality that exists outside of that experience in which everything at every scale is continuously changing.

A Dedekind cut of the reals gives (∞, 0) [0, ∞) or (∞, 0] (0, ∞), which are not exactly equal. I am not sure this has great impact on physics, unless there is a delta function set at 0. A course in real analysis covers these matters pretty extensively.

Grothendieck disappeared sometime back in the 1980s, and is suspected of living in S. France somewhere. The man is a bit of a subject from "A Beautiful Mind." During the Vietnam war he conducted lectures on algebraic geometry in the forests outside Hanoi as B-52s were bombing the place. I am not sure if he had much of an audience. In fact as I understand he is not a citizen of any country. He was fairly notorious for strange behavior. I think getting a hold of the man is a bit like trying to put a neutrino in a box. Connes' book on noncommutative geometry covers some aspects of Grothendieck's fibration system.

Cheers LC

Eckard Blumschein,

You said "If we agree that real numbers are different from rational ones then Georgina is wrong: The distinction between open and closed sets is invalid for really real numbers. oo oo = oo at least according to the really great ones like Galilei."

Please could you elaborate on why this should be so? I know nothing about this topos mathematics other than what I have read on this site and Wikipedia. The open set just sounded a good description that might fit with what I am trying to explain, to the best of my limited ability.

From Wikipedia..

A set U is open if any point x in U can be moved by a small amount in any direction and still be in the set U. The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Concepts that use notions of nearness, such as the continuity of functions, can be translated into the language open sets.

I was thinking that U could be that thing that we consider a particle and x could be the elusive dynamic entity squiggling about within set U with 4 degrees of freedom. So the set is made from all the squiggling. I am not a mathematician, so please forgive my language.

I am contemplating something that exists only because of its change in position in 4 dimensions. A component of that change in position (4th dimensional change) is not directly countable and could be envisioned being along a real number line. (Real numbers being used to measure continuous quantities.) It can be hypothesised that this is the spin and aforeward progression along the 4th dimension of a particle (except for antimatter which may progress aftward). The entity itself is counted as 1 although its continuous change means that it is not the identical 1 but a series of alternative 1s.

Its existence in our universe depends upon its 4th dimensional change in position, a real number continuous change.It can not be directly measured, we can only use time as a proxy and approximation. The 1 that is observed "frozen in an instant of existence" is then a function of being considered singular entity in 3D space denoted by the rational number 1 and having an unmeasurable real number change in position along the 4th spatio-energetic dimension. What does that make it? The entity can be considered as having both a rational component, if its changing is ignored and it is considered just 1 entity, and a real component because of its continuous change.

I think the interesting problem is with saying it is 1 when it is just a "snapshot" of the manifestation of something that is in a process of continuous change.

Last post was me. These numbers are annoying me.

The rational number assigned to an entity is just an integer quantity. Giving how many of it there are. But a physical entity is not just an integer quantity. It must also be seen to be a continuous process of change that gives it existence. So it must therefore be represented by a series of real numbers, if it is to be properly comprehended, whilst still retaining its integer assignment.

I am not saying that the rational integers are different from the real numbers. The integers must form part of the full set of real numbers.

It is how they are applied within physics to an entity that is problematic when the real number (4th dimensional) change is not considered. There is then a series of the same integer applying to different spatial manefestations of an entity. Which is correct because there can still the same quantity of it, whatever it has become.

1 ice cube, 1 puddle, 1 vapour cloud. Or 1 electron, 1 positron or 1 infant, 1 child, 1 adult. Same quantity but also changing form.Bizarrely this means if 1 sheep turns into 1 banana it is still OK because there is still only 1 entity. There is still only 1 kind of 1, but a series of different manifestations of the 1 thing.I know I'm repeating myself and probably getting unnecessarily concerned about how this is described numerically but I want to be sure I have understood and clarified it reasonably.

Rational numbers are the ratio of two integers r = a/b. Now there are numbers which we call irrational, for they can't be expressed as such. A simple example is sqrt{2}. Assume it is a rational number sqrt{2} = a/b. Now this ratio must be something other than 2^n/2^m, for otherwise we would have sqrt{2} is some multiple of 2 or 1/2. Therefore b must be an odd number. Now we then have

sqrt{2}^2 = (a/b)^2,

or a^2 = 2b^2. Therefore a^2 is even and then must also be a. So a is a number a = 2c, and

2b^2 = 4c^2 ==> c^2 = b^2/2

so that b^2 is clearly even as well, and so them must be b. Yet b must be odd by above, so this is a contradiction. Therefore sqrt{2} can't be expressed as the ratio of two integers.

The great majority of numbers are irrational, and in fact they can't be counted. Galois algebra is a system for defining roots of polynomial equations, which are irrational, but the procedure describes a countably infinite set - not all of them. There are in the set of irrational numbers transcendental numbers, such as π. These numbers are not algebraic and they exist in a set which is not countably infinite. This then gets into the matter of infinite cardinalities, which in set theory results in the Cantor description of transfinite numbers and the continuum. The continuum hypothesis is a lynchpin conjecture on the Zermelo-Fraenkel set theory. Bernays and Cohen showed by using Godel's theorems that the continuum hypothesis is consistent with ZF, but not provable.

What you describe with open sets is a way of setting up Hausdorff point set topology. There are non-Hausdorff topologies as well, such as the moduli space for general relativity or Zariski topology that topos theory depends on. I am not sure how far to go in describing this stuff. This gets into advanced undergraduate to first year graduate school mathematics.

Cheers LC

Lawrence and Georgina,

Continuity vs. discreteness is an important issue in physics. Common sense distinguishes between what is considered as a number of countable indivisible items and what is modeled as an endlessly divisible and therefore uncountable liquid. Peirce defined a continuum as something every part of which has parts. Euclid called a point something that does not have parts. Accordingly, genuine continua like a volume, an area, or a line can just be approximated by a finite number of points, no matter how densely the points are thought to be arranged. Any finite set of points is zero-dimensional.

Do the natural numbers constitute a set? If we clarify that being infinite is a property, then we have to attribute this property to the rule of counting. The natural numbers are rather something to be set as large as desired. There is no largest number. The natural numbers are countable without restriction while the expression infinity means all of them and does not denote a quantum but something else, something uncountable.

From the possibility of bijection between 1, 2, 3, ... and 1, 4, 9, ... Galilei correctly concluded that the relations smaller, equal, or larger are not valid for (the entities of) infinite quantities but only for finite ones.

Georg Cantor started with the silly idea to count all numbers. When he used bijection to redefine countability and declared the natural numbers countably infinite, he did not accept that while any natural number is countable there is no countable entity of all natural numbers. Nonetheless, he correctly found out that every rational number is also countable.

In order to count uncountable numbers too, he introduced the most cardinal mistake into mathematics: Cardinality. Instead to simply distinguish discrete and accordingly countable numbers from uncountable real numbers, he draw the wrong conclusion that there must be more real numbers than natural ones. Common sense tells us that this does not work with the original meaning of being absolutely infinite. Cantor, whose mother was Catholic, tried in vain to convince cardinal Franzelin that there is an Infinitum creatum sive Transfinitum below the Infinitum aeternum increativum sive Absolutum. Up to now, the notion infinity is murky in set theory. In a letter to Dedekind, Cantor himself excluded what he called "absolut unendliche" alias "inkonsistente Vielheiten".

Dedekind's cut postulates - without any possibility for a proof - nothing else than trichotomy for the real numbers too: "... every point of one piece is located left from every point of the other one."

This is clearly at odds with the impossibility to resolve a line into a finite number of points. How differ so called real numbers from the rational ones? How can they fill the gaps between any two rational numbers?

Let me tell it as simply as possible: Rational numbers - irrational ones as well as incorporated rational ones - must be irreal in the sense that they are not countable. This means, they have to be represented like p/q with p as well as q infinite, in other words with infinite precision. Otherwise, for instance pi would differ from exact value. This means, even the tiniest difference between two rational numbers must be filled with a continuum which is not yet adequately described by the hugest number of points. Continuity is a different quality.

Imagine a letter with an uncountably lengthy post code. It will never arrive. While the difference between subsequent rational numbers can be made as small as you like, the difference between subsequent real numbers must be absolutely zero.

Of course, only approximations to such really real numbers can be numerically handled. On the other hand, several apparent problems including Schroedinger's cat vanish when we do not forget that mathematics is always treating the reals as if they were rationals. Weyl called them aptly a sauce.

Forget 0)[0 and 0](0 and 0)0(0 and the possibility to choose at will. The correct solution is to understand: For really reals, there is no difference between [ and ( at all.

Goedel questioned the hypothesis c = aleph_1 = 2 ^(aleph_0). He considered c much larger. So far there is no tenable basis for a quantitative comparison between the two qualities.

Regards,

Eckard

I think you might want to read up on real variables, or take a course on the subject. With an MS in math I took a graduate level course, but frankly the subject is not exactly my cup of tea in mathematics. I find it tedious and not terribly interesting. Yet, issues of open and closed sets are well defined, including lim_[sup} and lim_{inf} of sets which approach sets (both open and closed) in certain Cauchy-like sequences. This is fairly classical mathematics, mid to late 19th century stuff. It all leads to generalized theory of integration such as Fatou's theorem and the Lebesgue integral.

With physics information is discrete. Even in classical mechanics. For a wave a discontinuities in a wave front will define a phase velocity for the flow of information. We might think of this with regards to Dirac's comb, or a discrete set of pulses. What is continuous does not convey information. This geometric aspect of theory permits us to work with differential and integral equations. This is IMO a great confusion with the Planck scale. This is the scale where the area of a black hole horizon is determined by a deBroglie wave, leading to L_p = sqrt{Għ/c^3} as some smallest region where physics obtains. Many people think this is some type of discretizing of spacetime. This is wrong, for all it tells us is the minimal scale where we can measure information about physics.

I will largely avoid the Zermelo-Fraenkel set theory issues. The subject captured my attention a number of years ago, and went through a lot of this. I can't really regurgitate much of it on a short blog post. I will say that largely I found the whole thing consistent and that is worked. I did not find it of much value for physics, so I have largely not studied this subject in over 10 years. The set-theory mavens did come up with an interesting proof in the 80s on the set of homeomorphic 4-manifolds and 7-spheres that are not diffeomorphic. So the subject is not without merits. These guys also work to make tests on how certain areas of mathematics have axiomatic structure consistent with ZF set theory.

Lawrence B. Crowell

Lawrence and Eckard,

Thank you for your explanations.I probably should think more about numbers.

It seems to me that the 4th spatio-energetic dimension must qualify as really representable by real numbers because there is no way to count or measure along it but if it is spatial then it must have distances. We can only use a continuous measurement undertaken in 3D space to represent that measurement (time).

I also think that even the concept of a point is problematic because to identify it the universe must be artificially frozen. If not only energy but spatial change is conserved then the whole universe is engaged in continuous change of quaternion spatial position. So a single point does not have an unchanging position in quaternion space. So the concept of a point is a mathematical abstraction rather than an accurate representation of physical reality.

At the very least any point with existence rather than a purely mathematical abstraction is moving along the 4th spatial dimension and therefore is rotating in 3D space. If it is a point on a piece of matter such as a page there is the rotation of the earth, orbit of the sun and rotation of the galaxy to consider. Those gimbals turn and the point, although it can be identified as 1 single point in space, also has a whole sequence of different spatial position due to 4 degrees of freedom of movement within space, which are unobserved and unmeasurable and it would seem to me must be considered as representable by real numbers.

Eckard Blumschein You said "Euclid called a point something that does not have parts." But is that really so if the point also ought to be described by a real sequence? So there is it seems to me a difference between the integer 1 and a single existential entity and between a point in mathematics and a point within quaternion energy-space or if you must think in terms of time within space-time. There is a difference between physics and mathematics.

Eckard, you said "Euclid called a point something that does not have parts. Accordingly, genuine continua like a volume, an area, or a line can just be approximated by a finite number of points, no matter how densely the points are thought to be arranged. Any finite set of points is zero-dimensional."

In existential physics however it would seem that no point can be zero dimensional because of the continuous change of spatial position of that point, with 4 degrees of freedom.(Even though it may appear stationary) Therefore no set of points within existential physics can be zero dimensional.

I am only thinking about what numbers -are- because it seems relevant to the current contemplation of how they are used for modelling in physics. I have no love for them nor them for me. (I am now re-acquainted with the various terms for different kinds of numbers, thank you.) It seems to me, following on from my previous posts, that the only way to accurately model even a point or single entity is to use a series of quaternion numbers.

Eckard you said "Of course, only approximations to such really real numbers can be numerically handled. On the other hand, several apparent problems including Schroedinger's cat vanish when we do not forget that mathematics is always treating the reals as if they were rationals. Weyl called them aptly a sauce."

I think that is really interesting. I like the analogy of sauce very much too.

from me.