REASONABLE
EFFECTIVENESS OF MATHEMATICS
INTRODUCTORY
The
validity of a mathematical statement is judged by its logical consistency. The
validity of a physical statement is judged by its correspondence to reality. We collect too
much data and without judging properly reject most (like at LHC). If we re-envision classical and quantum observations as macroscopic overlap
of quantum effects, we
may solve most problems. The physics community
blindly accepts rigid, linear ideas about the nature of space, time, dimension,
etc. These theories provide conceptual convenience and attractive simplicity
for pattern analysis, but at the cost of ignoring equally-plausible alternative
interpretations of observed phenomena that could possibly have explained the universe
better. Modern theories do not give a precise definition of the technical terms
used, but give an operational definition that can be manipulated according to
convenience. Wigner1 defined mathematics as the science of skillful
operations with concepts and rules invented just for this purpose. This is too
open-ended. What is skillful operation? What are the concepts and Rules? Who
invented them? What is the purpose? Do all concepts and rules have to be
mathematical? Wigner says: The great
mathematician fully, almost ruthlessly, exploits the domain of permissible
reasoning and skirts the impermissible, but leaves out what is permissible
and what is not; leaving scope for manipulation.
Relations
between material objects must be expressed in a language compatible with the
way in which objects in the real world actually interact - through the
transmission/reception of mass/energy/information. Every object is a summation of the same fundamental stuff (quarks,
leptons, etc) in varying orders. Events are energy rearranging fundamental
particles. The space-time location makes intervals in both space and
time dependent on where we measure them from. This implies space-time is
related to the origin of the coordinates of the observer’s frame of reference.
Measurement is carried out at here-now – thus, time variant (since now is the
fleeting interface between past and future). Its quantitative description is
mathematics – it describes the changing physical phenomena when the number or
arrangement of any of the constituent parameters is changed. The changes are expressed as the result of
measurement after comparison with a scaling constant (standard unit). These are
always pure numbers, i.e., scalar quantities, because measurement is only the
operation of scaling up or down the unit for an appropriate number of times. The
results of measurement, which are time invariant, are frozen even though the
object measured continues to evolve in time. Your 10 year old photo is not you.
Mathematics is the ordered accumulation
and reduction in numbers of the same class (linear or vector) or partially
similar class (non-linear or set) of objects. Number is one of the properties of all substances by which we
differentiate between similars. If there is nothing similar at here-now, the
number associated with the object is one. If there are similars, the
number is many. Our sense organs and measuring instruments are capable
of measuring only one at a time. Thus, many is a collection of
successive one’s. Based on the sequence of perception of such one’s,
many can be 2, 3, 4….n. In a fraction, the denominator represents the one’s,
out of which some (numerator) are taken. Zero
is the absence of something at here-now that is known to exist elsewhere
(otherwise we will not perceive its absence at all).
Burrowing
from M. Polanyi, Wigner says1: The
principal point …. is that the mathematician could formulate only a handful of
interesting theorems without defining concepts beyond those contained in the
axioms and that the concepts outside those contained in the axioms are defined
with a view of permitting ingenious logical operations which appeal to our
aesthetic sense both as operations and also in their results of great
generality and simplicity. Wigner admits not only the incompleteness of
mathematics but also its manipulation according to the aesthetic sense of the operator. He gives the example of complex
numbers and burrowing from Hilbert2, admits: Certainly, nothing in our experience suggests the introduction of these
quantities. Indeed, if a mathematician is asked to justify his interest in complex
numbers, he will point, with some indignation, to the many beautiful theorems
in the theory of equations, of power series, and of analytic functions in
general, which owe their origin to the introduction of complex numbers. The
mathematician is not willing to give up his interest in these most beautiful
accomplishments of his genius. A reverse self-fulfilling effect!
Negative numbers are related to mutually exclusive
objects or events of a coupled system. For example,
position (fixed coordinates) and momentum (mobile coordinates) are mutually
exclusive. In two accelerating frames of reference, one who gains has positive
value corresponding to the negative value of the other. Since one is
without similars, it does not change the value in any operation except linear
addition and subtraction (becoming many or zero). Thus, squaring or square-root
of 1 is 1 (these involve field). Since negative numbers belong to mutually
exclusive couplets and not exclusive one’s, complex numbers are neither
physically nor mathematically valid. No
computer algorithm is possible using complex numbers.
Infinity is
like one: without similars. But whereas the dimensions of one are fully
perceptible, the dimensions of infinity
are not perceptible. There cannot be negative infinity to positive infinity
through zero, as it will show one beginning or end of infinity at the zero
point, which is non-existent at here-now. No mathematics is possible with
infinity, as all operations involving it will have undefined dimensions – thus
indistinguishable from each other. History shows that whenever infinity appears
in any theoretical model, it points to some fundamentally different and novel
phenomena. In aerodynamics formulas, as the velocity approached the velocity of
sound in the medium where the aircraft moved, the resistance of the medium
returned infinite figure. It was believed that supersonic motion is impossible.
But when supersonic motion became obvious, the formulas were reviewed. It was
noted that they described resistance only in a continuous medium without abrupt
jumps in density and pressure. However, transition from subsonic to supersonic
motion involves a shock wave in front of the body, leading to abrupt jumps in
density and pressure. When these factors were taken into account, the infinity
vanished.
The so-called
irrational numbers are also perceived as the nearest fraction of integers.
Otherwise, we cannot use them in programming. We may be as precise as we want
to fix the value of a number tending to zero, but it will never be zero, as
that will make it non-existent at here-now making the operation impossible.
Similarly, a number tending to infinity will never become infinite, as the
result of all such operations become indistinguishable from each other. Like
energy, infinities coexist. Only, space, time, coordinates and consciousness
are infinite.
Language
is the transposition of information to another system’s
CPU or mind by signals or sounds using energy (self communication is
perception). The transposition may relate to a fixed object/information.
It can be used in different domains and different contexts or require
modifications in prescribed manner depending upon the context. Since
mathematics follows these rules, it is also a language. Mathematics explains only how much one
quantity, whether scalar or vector; accumulate or reduce linearly or
non-linearly in interactions involving similar or partly similar quantities and
not what, why, when, where, or with whom
about the objects. These are subject matters of physics. The interactions are
chemistry. There is no equation for Observer. The enchanting smile on the lips
of the beloved is not the same as geometry of mouth or curvature of lips. Thus,
mathematics is not the sole language of Nature.
MATHEMATICAL
PHYSICS
Because
of logical consistency, mathematics is always deterministic. Look at the
structure of any equation. The initial condition or parameters are represented
by the left hand side. The equality sign describes the special conditions to be
met to start any interaction: be it at macro level or micro level. Given the
initial conditions, the right hand side describes the theorized outcome of the
interaction. We are free to vary the parameters of the left hand side. That is
our freewill (though our choices or degrees of freedom may be variously limited).
Once the initial parameters are set; (math can’t predict this), the right hand
side (final outcome), varies correspondingly. This predetermined outcome is
mathematics. The equality signs - the special conditions (like temperature
threshold to start a chemical reaction), are also predetermined. But it is not
defined in a logically consistent way (why that temperature?) – hence not
mathematical.
Some
say; mathematics, because of its inbuilt logic, writes itself - one can start writing
things down without knowing exactly what they are, and the language makes
suggestions to proceed. This is the ergodic monkey phenomenon, where a monkey
plays with the key-board randomly and the outcome is a master piece of a novel.
Though theoretically it is possible as a chance, it does not happen in reality.
Others say: Master enough of the basics;
and you rapidly enter what sports players call ‘the zone’. Suddenly it gets
much easier. You are propelled along. This is the 100th monkey phenomenon
of Sheldrake - notion that new skills are learnt with increasing ease as
greater quantities of a population acquire them. There is no proof to justify
this view beyond chance and functional ease due to repeated use.
Wigner
says1: applied mathematics is
not so much the master of the function: it is merely serving as a tool. Others say; using mathematics, we can build abstract
models without the restrictions imposed by the physical world. This leads
to the incompleteness issues, which exploit problems arising out of unnatural
mathematics. We see something when the radiation emitted by it interacts with
our eyes. We touch the mass that radiates light. Thus we do not touch what we
see (radiation) and see what we touch (mass). Nature prohibits reductionism. Whole is a sum of its parts and
more. Water is more than 2H and O. A triangle is more than three straight
lines. This is natural number theory. 5
has independent perceptual value than 5 ones. If we can purchase a car in € 5k, with € 1k,
we can purchase 1/5 of a car. This may look
mathematically valid, but 1/5 of a car
is an undecidable proposition. Hilbert’s
problem whether mathematics is complete
(every statement in the language of number theory can be either proved or
disproved) and Gödel’s negative solution arise out of such unnatural mathematics.
Brute force approach is similarly unnatural, though sometimes it may succeed by
chance.
Wigner
is right when he talks about1 the
succession of layers of laws of nature, each layer containing more general and
more encompassing laws than the previous one and its discovery constituting a
deeper penetration into the structure of the universe than the layers
recognized before. This is the principle that both macrocosm and microcosm
replicate each other. As the Minutes of the American Mathematical Society for
October, 2005 reported, the theory of dynamical systems used to plan
trajectories of spacecrafts and those of transition states of chemical
reactions share the same set of mathematics. Wigner is also right that all these laws of nature contain … only a
small part of our knowledge of the inanimate world. But he misses the point
when he says: All the laws of nature are
conditional statements which permit a prediction of some future events on the
basis of the knowledge of the present, except that some aspects of the present
state of the world…are irrelevant from the point of view of the prediction.
In fact, it is most relevant as the probabilistic laws of Nature. The
conditional statements show interdependence of all systems in the cosmos. Our
sense organs and measuring devices have limited capacity, so that it measures
limited aspects in limited intervals. Since time evolution is not uniform, but
conditional on interactions, we do not see each step from the flapping of the
wings of the butterfly till it turns into tempest elsewhere. The creation is highly ordered and there is
no randomness or chaos. We fault
Nature to hide our inability to know.
Wigner
says: The physicist is interested in
discovering the laws of inanimate nature…. It is, as Schrodinger has remarked,
a miracle that in spite of the baffling complexity of the world, certain
regularities in the events could be discovered. In an earlier paper3,
we have shown that: uncertainty is not a
law of Nature. It is the result of
natural laws relating to measurement related to causality that reveal a
kind of granularity at certain levels of existence. The uncertainty relation of
Heisenberg was reformulated in
terms of standard deviations, where the focus was exclusively on the
indeterminacy of predictions, whereas the unavoidable disturbance in
measurement process was ignored. A formulation of the error – disturbance
uncertainty relation, taking the perturbation into account, was essential for a
deeper understanding of the uncertainty principle. By directly measuring errors and disturbances in the observation of spin
components, Ozawa developed a formulation: ε (q) η(p)
+ σ(q)η(p) + σ(p)ε(q) ≥
h/4π. Ozawa’s inequality
suggests that suppression of fluctuations is not the only way to reduce error,
but it can be achieved by allowing
a system to have larger fluctuations. Nature Physics
(doi:10.1038/nphys2194) describes a neutron-optical experiment that records the
error of a spin-component measurement as well as the disturbance caused on
another spin-component. The results confirm that both error and disturbance
obey the new relation but violate the old one in a wide range of experimental
parameters. Even when either the source
of error or disturbance is held to nearly zero; the other remains finite: thus,
mutually exclusive. that is said to encode the causal structure of
Spacetime. Each event in Spacetime has a double-cone attached to it, where the
vertex corresponds to the event itself. Time runs vertically - the upward cone opens to future of this
event. The downward cone
shows past.
But if the light pulse radiates in all directions, it should show concentric spheres and not a double-cone. The trick
is done by first taking two dimensions and time as the third dimension. But
even then it will be concentric circles and not a conic section. Event horizon is the limit of our vision.
Light Cone
is a mathematical model
Not
only time is cyclic, but also is unidirectional because ‘now’ is linked to future in a different way; than it is linked to the
past. Space, Time and coordinates arise from our concept of sequence and
interval. When it is related to objects, we call the interval space. When it is
related to events, we call the interval time. When we describe
inter-relationship of objects, we describe the interval by coordinates. Present and future are segments of these sequences of intervals that are
strictly ordered - future always follows present. The same is not true for
past, because any past event can be linked to the present bypassing the specific
sequence. This proves unidirectional time. Since the intervals are infinite,
space and time are an infinite continuum. We use segments of this analog
reality. Thus, our description relates to here-now over a medium scale, which
we tend to universalize. This gives a distorted picture.
MISSING THE WOODS
FOR THE TREES
Does the structure and availability of
existing mathematics shape the formulation of physical theories? Yes! Examples:
Dimension
is the perception of differentiation between internal structural space and
external relational space of an object. Since we observe through electromagnetic
radiation, where the electric field and the magnetic field move perpendicular
to each other and both move perpendicular to the direction of motion, we have
three mutually perpendicular dimensions representing length, breadth, height that are invariant under mutual
transformation.
There are no extra large or compact or
n’th dimension. The surface of a cube is not 2D, as it has no independent
existence. There is no independent straight line in 1D. It is a mark on a three
dimensional paper or space. The surface of a sphere is not 1D, but 3D.
Directions (axes) and sequential arrangement (coordinates) of an object are used with reference to an
origin in relation to other objects. With only one object, direction is
meaningless. Direction is used to:
1) Measure distance between two objects from origin by assigning + or –
signs from origin along various axes.
2) Indicate shortest distance between two objects on a curved surface
like a geodesic.
3) Reflect the behavior of fundamental forces of Nature; i.e., strong
interactions move towards center, part of weak interactions limits movement away from
center, e.m. interaction move out from center to lower concentration, beta-decay separates a part from the center, gravitational interaction relates interaction between bodies. But often, dimension is exchanged for direction
in phase space portrait and its quantum Avatar, Hilbert space. This leads
to the undecidable
propositions.
If we divide 20
by 5, then we take out bunches of 5 from the lot of 20. When the lot becomes
empty or the remainder is zero or below 5 (divisor) so that it cannot be
considered a bunch and taken away further, the number of bunches of 5 are
counted. That gives the result of division as 4. In case of division by zero,
we are supposed to take out bunches of zero, which is impossible as it is not
at here-now of the operator. At no stage the lot becomes zero or less than
zero. Thus, the operation is not complete and result of division cannot be
known, just like while dividing 20 by 5, we cannot start counting the result
after taking away two bunches. Conclusion: division by zero is mathematically
void; hence it leaves the number unchanged. Since zero does not exist at
here-now, it does not affect addition or subtraction. During multiplication by
zero, one non-linear component of the quantity is extended to zero, i.e., moves
away from here-now to a superposition of states. Thus, the result becomes zero
for the total component, as we cannot have a Schrödinger’s undead cat in real life.
In division by zero, the non-existent part is sought to be reduced from the
quantity (which is an operation akin to collapse reversal in quantum
mechanics), leaving the quantity unchanged. Division by zero leaves the
number unchanged.
Two possibilities of measurement of a moving rod suggested by Einstein in his 1905 paper4
were:
(a) The observer moves together with the given measuring-rod and the rod to
be measured, and measures the length of the rod directly by superposing the
measuring-rod, in just the same way as if all three were at rest, or
(b) By means of stationary clocks set up
in the stationary system and synchronizing with a clock in the moving frame,
the observer ascertains at what points of the stationary system the two ends of
the rod to be measured are located at a definite time. The distance between
these two points, measured by the measuring-rod already employed, which in this
case is at rest, is the length of the rod.
The method described at (b) is
misleading. We can do this only by setting up a measuring device to record the
emissions from both ends of the rod at the designated time, (which is the same
as taking a photograph of the moving rod) and then measure the distance between
the two points on the recording device in any unit. But the picture will not give a correct reading because:
- If the length
of the rod is small or velocity is small, then length contraction will not
be perceptible according to the formula given by Einstein.
- If the length
of the rod is big or velocity is comparable to that of light, then light
from different points of the rod will take different times to reach the
recording device and the picture we get will be distorted due to different
Doppler shift. Thus, there is only one way of measuring the length of the
rod as in (a).
The fallacy in the above description is
that if one treats as if all three were
at rest, one cannot measure velocity or momentum, as the object will have zero
relative velocity. Einstein missed this point when in the same paper4,
he said: Now to the origin of one of the
two systems (k) let a constant
velocity v be imparted in the
direction of the increasing x
of the other stationary system (K), and let this velocity be communicated to
the axes of the co-ordinates, the relevant measuring-rod, and the clocks. But
is this the velocity of k as measured from k, or is it the velocity as measured
from K? K and k each have their own clocks and measuring rods, which are not
treated as equivalent by Einstein. Therefore, according to his theory, they
will measure the velocity of k differently. Einstein does not assign the
velocity specifically to either system. Everyone missed it and got misled. His
spinning disk example in GR also falls for the same reason.
On the
definition of synchronization Einstein says: Let a ray of light start at the
“A time” tA from A towards B, let it at the “B time” tB
be reflected at B in the direction of A, and arrive again at A at the “A time”
t’A. In accordance with definition the two clocks synchronize if:
tB - tA = t’A - tB.
We
assume that this definition of synchronism is free from contradictions, and
possible for any number of points; and that the following relations are
universally valid:
- If the clock
at B synchronizes with the clock at A, the clock at A synchronizes with
the clock at B.
- If the clock
at A synchronizes with the clock at B and also with the clock at C, the
clocks at B and C also synchronize with each other.
The concept of
relativity is valid only between two objects. Introduction of a third object
brings in the concept of privileged frame of reference and all equations of
relativity fall. In the above description, the
clock at A is treated as a privileged frame of reference for proving
synchronization of the clocks at B and C. Yet, he claims it is relative!
Russell’s paradox raises an interesting question: If S is
the set of all sets which do not have themselves as a member, is S a member of
itself? The general principle is that: there cannot be a set without individual
elements. Collection of different objects unrelated to each other would be
individual members as it does not satisfy the condition of a set. Thus a
collection of objects is either a set with its elements, or individual objects
that are not the elements of a set.
Example: p(x): x Ï x, is the defining property p(x) of any
element x such that it does not belong to x. Many sets have this property. A
library p(x) is a collection of books. But a book is not a library x Ï x. Suppose this property defines the set R ={x : x Ï x}. It must be possible to determine if RÎR or RÏR. However if RÎR; then the defining properties of R implies RÏR, which contradicts the supposition that: RÎR. Similarly, the supposition RÏR confers on R the right to be an element
of R, again leading to a contradiction. The only possible conclusion is that,
the property x Ï x cannot define a set. It is convenient to
choose a largest set in any given context called the universal set and
confine the study to the elements of such universal set only. This set may vary
in different contexts, but in a given set up, the universal set should be so
specified that no occasion arises ever to digress from it. Otherwise, there is
every danger of colliding with paradoxes such as the Russell’s paradox. In the
case of EP, we do blunder!
All objects fall in similar ways under the
influence of gravity. Hence locally, it is said, the difference between an
accelerated frame and an un-accelerated frame cannot be known. But these must be related to
be compared as equivalent or not? In the
example of a person in an
elevator falling down a shaft, it is assumed that during any sufficiently small amount of time or over
a sufficiently
small space, the person can make no distinction between
being in the falling elevator and being stationary in completely empty space,
where there is no gravity. This is a wrong description – distinction of what? Unless
we relate the elevator to the outside space, we cannot relate motion of the elevator to it. The moment
we relate to the structures beyond the elevator, we can know the relative
motion of the elevator.
Inside a spaceship
in deep space, objects behave like Brownian motion (unaccelerated) or like the
asteroids in the asteroid belt (accelerated). Usually, they are relatively stationary within the medium unless
some other force acts upon them. If the person can see the outside
objects, then he can know the relative motions by comparing objects at
different distances. If he cannot see the outside objects, then he
will consider only his position with reference to the spaceship – stationary or floating
within a frame. There is no equivalence
because there is no other frame for comparison. Relativity theory needs
revision.
A same logic applies
to the ray of light that appears curved to
the occupants of the spaceship. The light can be related to the spaceship only
if we consider the bigger frame of reference containing the source of light and
the spaceship. If we consider outside space as a separate frame of reference
unrelated to the spaceship, the ray emitted by it cannot be considered inside
it. If the passenger could observe the scene outside, he will notice this
difference and know that the spaceship is moving. Otherwise, the consideration
will be restricted to the rays emanating from within, which will move straight.
In either case, the description is faulty. Thus, the foundation of GR - the EP
- is wrong description of reality.
Hence all mathematical derivatives built upon such wrong description are wrong.
There is no inertial mass increase.
Einstein
has used equations x2+y2+z2- (ct)2
= 0 and ξ2 + η2 + ζ2 – (cτ)2
= 0 to describe the evolution of the same light pulse that the observers see. But
x2+y2- (ct)2 = 0 describes a circle with ct as
the radius! Hence z and ζ have to be zero. It can’t be a sphere! Since (x. y. z) is a point on the
circumference, moving in z direction will be tangential. It describes a cylinder
and not a sphere! The geometer’s
descriptions of π-sphere and the topologist’s descriptions of n-dimensional
sphere are mathematically and physically void.
Einstein
can describe two concentric spheres with the points (x,y,z) and (ξ, η, ζ ) on
their respective circumferences. Since the second person is moving away from
the origin, the second equation relates to sighting from his here-now (new origin).
Assuming he sees the same sphere, he should know its origin (because he has
already seen it, otherwise he will not know that it is the same light pulse).
In case he takes a new measurement from his origin, according to Einstein, the
reading from two frames will be different. In other words, he will either
measure it independently as different or measure the same radius as the other,
implying: either:
x2+y2+z2-
c2t2 ≠ x’2+y’2+z’2-c2
τ 2, t ≠ τ.
Or
c2t2 = c2 τ2
or t =
τ. This creates
contradictions, which invalidates his mathematics.
LOOKING AHEAD
Recent
discovery of galactic blue-shift (Lowell
Observatory Bulletin No. 58 Vol. II No. 8) and arXiv: 1402.6319 v1 [astro-ph.GA], galactic merger (Astronomy Newsletter
- 18th July, 2014) and the absence of red-shift in galactic
or lesser scales should prove dark energy a myth. Energy is perceived through
its interactions. If it is not interacting, it cannot be energy. Fluids are
also smooth and persistent. Interpretation of M & M experiment is faulty,
as light is a transverse wave, which is background invariant. Like the solar
system, the universe is spinning around a galactic center. Dark energy is the
universal background structure.
Abundance
of ‘Hot Jupiters’ among alien planets and protons in cosmic rays shows
macro-micro relation. Separately we have shown that the internal structures of
Jupiter and proton are similar.
Bare mass
or bare charge is fiction. The equation e = mc2 is mathematically
invalid as LHS is time invariant and RHS time variant (per second). It should
be written as (m) e → (mc)2 which balances it to show the rate at
which energy acts on mass. Energy cannot be confined in packets, but only by
mass. Confined mass-energy is fermion (hence half integer spin) and unconfined
mass-energy is boson (zero spin). Fluid (intermediate) behavior is integer
spin.
Result
of Time dilation experiment with atomic clocks was faked. This can be verified
from the original data kept at US Naval Archives. The delayed signal of GPS is due
to refraction when the signal re-enters the denser atmosphere of Earth. Time
dilation is relative time evolution of elements in entropy, where thermodynamic
process sustains life and total disorder is annihilation of form.
There is a need to ponder over these
issues, introspect and rewrite physics.
REFERENCES
1.
Eugene
Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural
Sciences,” in Communications in Pure and Applied Mathematics, vol. 13, No. I
2.
Abh.
Math. Sem., Univ. Hamburg, 157 (1922), or Gesammelte Werke (Berlin: Springer,
1935), p. 188.
3.
“Is Reality Digital or Analogue” published by
the FQXi Community on Dec. 29, 2010,
4.
“On
the Electrodynamics of Moving Bodies” by Einstein.